# Dynamical (In)Stability of Ricci-flat ALE metrics along the Ricci flow

Alix Deruelle  Institut de mathématiques de Jussieu, 4, place Jussieu, Boite
Courrier 247 - 75252 Paris<EMAIL_ADDRESS>and  Tristan Ozuch  MIT,
Dept. of Math., 77 Massachusetts Avenue, Cambridge, MA 02139-4307.
<EMAIL_ADDRESS>
(Date: today)

###### Abstract.

We study the stability and instability of ALE Ricci-flat metrics around which
a Łojasiewicz inequality is satisfied for a variation of Perelman’s $\lambda$
functional adapted to the ALE situation and denoted
$\lambda_{\operatorname{ALE}}$. This functional was introduced by the authors
in a recent work and it has been proven that it satisfies a good enough
Łojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat
metrics in dimension larger than or equal to $5$.

###### Contents

  1. 1 The functional $\lambda_{\operatorname{ALE}}$ and its properties
    1. 1.1 First definitions and function spaces
    2. 1.2 The functionals $\lambda_{\operatorname{ALE}}^{0}$, $m_{\operatorname{ADM}}$ and $\lambda_{\operatorname{ALE}}$
      1. 1.2.1 Mass of ALE metrics
      2. 1.2.2 The functional $\lambda_{\operatorname{ALE}}^{0}$
      3. 1.2.3 Definition of the functional $\lambda_{\operatorname{ALE}}$
    3. 1.3 Main properties of $\lambda_{\operatorname{ALE}}$
      1. 1.3.1 Variations of $\lambda_{\operatorname{ALE}}$
      2. 1.3.2 A Łojasiewicz inequality
  2. 2 Preliminary estimates for short and large time
    1. 2.1 Heat kernel Gaussian bounds
    2. 2.2 $L^{2}-C^{0}$ estimate
    3. 2.3 Weighted estimates
  3. 3 Stability of Ricci-flat ALE metrics
    1. 3.1 A stability result
    2. 3.2 Proof of Theorem 3.1
    3. 3.3 Evolution of the Bianchi form, the scalar curvature and the mass
    4. 3.4 A discussion on previous stability results
  4. 4 Instability of Ricci-flat ALE metrics
    1. 4.1 A priori energy estimates in the unstable case
    2. 4.2 A digression on $\lambda_{\operatorname{ALE}}$
    3. 4.3 An unstability result
  5. A First and second variations of geometric quantities

## Introduction

The understanding of Ricci-flat ALE metrics is a central issue in Riemannian
geometry. These spaces model the formation of singularities of spaces with
Ricci curvature bounds [And90, BKN89] as well as the singularities of
$4$-dimensional Ricci flow with bounded scalar curvature [BZ17], [Sim20]. They
moreover appear as finite-time blow-up limits of some Ricci flows [App19].
Their stability therefore becomes a crucial question for the Ricci flow.

Our goal here is to study the _dynamical_ stability and instability of these
spaces along the Ricci flow thanks to a functional on suitable neighborhoods
of any ALE Ricci-flat metrics which detects Ricci-flat metrics as its critical
points.

### An adaptation of Perelman’s $\lambda$-functional to the ALE setting

In [DO20], the authors introduced an adaptation to the ALE setting of the
$\lambda$-functional from [Per02], this functional is denoted
$\lambda_{\operatorname{ALE}}$. More precisely, for any ALE metric $g$ of
order $\tau>\frac{n-2}{2}$ (see Definition 1.1) whose scalar curvature is
integrable, we define

$\lambda_{\operatorname{ALE}}(g):=\lambda^{0}_{\operatorname{ALE}}(g)-m_{\operatorname{ADM}}(g),$

where
$\lambda^{0}_{\operatorname{ALE}}(g):=\inf_{w}\int_{N}4|\nabla^{g}w|_{g}^{2}+\mathop{\rm
R}\nolimits_{g}w^{2}$ where the infimum is taken among the smooth functions
$w$ with $w-1$ compactly supported, and $m_{\operatorname{ADM}}$ is (up to a
constant) the ADM mass of $(N,g)$. The point of substracting
$m_{\operatorname{ADM}}(g)$ is that the functional now extends to an analytic
function on the classical space of metrics satisfying, for
$\tau>\frac{n-2}{2}$, $k\in\\{0,1,2,3\\}$ and a Ricci-flat metric $g_{b}$,
$(1+r)^{\tau+k}|\nabla^{g_{b},k}(g-g_{b})|\ll 1$ where $r$ is the
$g_{b}$-distance to a given point. Neither $\lambda^{0}_{\operatorname{ALE}}$
or $m_{\operatorname{ADM}}$ can be defined on such a neighborhood without
further constraints.

In [DO20], we moreover showed that $\lambda_{\operatorname{ALE}}$ detects
Ricci-flat ALE metrics as its only critical points, and the Ricci flow is
moreover its gradient flow.

The second variation of $\lambda_{\operatorname{ALE}}$ at an ALE Ricci-flat
metric $(N^{n},g_{b})$ along divergence-free variations is half the
Lichnerowicz operator $L_{g_{b}}:=\Delta_{g_{b}}+2\mathop{\rm
Rm}\nolimits(g_{b})\ast$. This leads us to define the linear stability of an
ALE Ricci-flat metric $(N^{n},g_{b})$ as the non-positivity of the associated
Lichnerowicz operator $L_{g_{b}}$ restricted to divergence-free variations. In
the integrable case, i.e. in the case where the space of ALE Ricci-flat
metrics in the neighborhood of a fixed ALE Ricci-flat metric is a smooth
finite-dimensional manifold, we have a nice consequence of the linear
stability: any linearly stable and integrable ALE Ricci-flat metric is a local
maximum for the functional $\lambda_{\operatorname{ALE}}$. Properties of the
ADM mass were deduced thanks to this fact in [DO20].

A more delicate notion of stability is that of _dynamical stability_ of ALE
Ricci-flat metrics along the Ricci flow. Its study on _compact_ Ricci flat
manifolds has been investigated in [GIK02], [Ses06], [Has12] and [HM14]. In
the non-compact situation, there are several additional difficulties. A major
difference is that $0$ is not isolated in the spectrum of the linearized
operator. This prevents an exponential convergence rate as in the case of
[Has12] of an integrable Ricci-flat metric on a closed manifold, one only gets
a polynomial-in-time convergence.

### A weighted Łojasiewicz inequality for $\lambda_{\operatorname{ALE}}$.

One tool that has been quite popular to study the dynamical stability of fixed
points of geometric evolution equations is the notion of Łojasiewicz-Simon
inequalities. Its name comes from both the classical work of Łojasiewicz
[Ło65] on finite dimensional dynamical systems of gradient type and that of L.
Simon [Sim83] who extended systematically these inequalities to functionals
defined on infinite dimensional spaces. The main geometric applications
obtained in [Sim83] concern the uniqueness of tangent cones of isolated
singularities of minimal surfaces in Euclidean space together with the
uniqueness of tangent maps of minimizing harmonic maps with values into an
analytic closed Riemannian manifold. These geometric equations have the
advantage to be strongly elliptic. Notice that all these results do not hold
true if one drops the assumption on the analyticity of the data under
consideration.

In the compact setting, Łojasiewicz inequalities have been proved for
Perelman’s $\lambda$-functional in the neighborhood of _compact_ Ricci-flat
metrics and were the main tool to study the stability of Ricci-flat metrics in
[Has12] in the integrable case, and in [HM14] in the general case.

The main result in [DO20] is that the functional
$\lambda_{\operatorname{ALE}}$ satisfies a weighted Łojasiewicz inequality in
a neighborhood of any ALE Ricci-flat metric with respect to the topology of
weighted Hölder spaces $C^{2,\alpha}_{\tau}$, $\alpha\in(0,1)$, with
polynomial decay of rate $\tau\in(\frac{n-2}{2},n-2)$ (see Definition 1.2). It
is _weighted_ since it uses the $L^{2}_{\frac{n}{2}+1}(g_{b})$-norm which is
essentially $L^{2}((1+r)dv_{g_{b}})$ instead of $L^{2}(dv_{g_{b}})$. Roughly
speaking, the article [DO20] proves that there is some $\theta\in(0,1]$ such
that for any metric $g$ in a $C^{2,\alpha}_{\tau}$-neighborhood of a given ALE
Ricci flat metric, the following Łojasiewciz inequality holds:

$|\lambda_{\operatorname{ALE}}(g)|^{2-\theta}\leq
C\|\nabla\lambda_{\operatorname{ALE}}(g)\|_{L^{2}_{\frac{n}{2}+1}(g_{b})}^{2}.$
(0.1)

We refer the reader to Theorem 1.12 for a precise statement.

The fact that our spaces are non-compact induces quite a lot of new
difficulties. In particular, the spectrum of the Lichnerowicz operator is not
discrete anymore and $0$ belongs to the essential spectrum. This explains the
need of considering weighted Sobolev spaces different from $L^{2}$ for which
the differential of the gradient $\nabla\lambda_{\operatorname{ALE}}$ at a
Ricci-flat ALE metric is Fredholm. Theorem 1.12 gives an optimal
$L^{2}_{\frac{n}{2}+1}$-Łojasiewicz inequality with exponent $\theta=1$ in the
integrable situation. Nonetheless, it seems that inequality appears not to be
so useful regarding the study of dynamical stability of the Ricci flow near an
ALE Ricci flat metric. This is essentially due to the $L^{2}$-variational
structure of $\lambda_{\operatorname{ALE}}$. For this reason, in the setting
of $L^{2}$-perturbations, by interpolation, we obtain the following
Łojasiewicz inequality near any ALE Ricci flat linearly stable and integrable
metric in dimension greater than $4$. In particular, if $n\geq 5$, one has the
following $L^{2}$-Łojasiewicz inequality for integrable Ricci-flat ALE
metrics: for $\tau\in(\frac{n}{2},n-2)$ and
$0<\delta<\frac{2\tau-(n-2)}{2\tau-(n-4)}$, there exists $C>0$ such that for
all $g\in B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$,

$|\lambda_{\operatorname{ALE}}(g)|^{2-\theta_{L^{2}}}\leq
C\|\nabla\lambda_{\operatorname{ALE}}(g)\|_{L^{2}(g_{b})}^{2},\quad\theta_{L^{2}}:=2-\frac{1}{\delta}<\frac{2\tau-n}{2\tau-(n-2)}.$
(0.2)

Notice that we cannot reach the usual optimal $L^{2}$-Łojasiewicz exponent
$\theta_{L^{2}}=1$. This is consistent with the known fact that the DeTurck-
Ricci flow only converges polynomially fast for perturbations of the Euclidean
space: see for instance [SSS11] and [App18]. We also refer to the recent work
[KP20] on the dynamical stability of integrable ALE Ricci flat metrics
carrying a parallel spinor. Indeed, an exponent $\theta_{L^{2}}=1$ would imply
that the convergence is exponential.

### Stability and instability of Ricci-flat ALE metrics.

In the present article, we study the _dynamical_ stability or instability of
Ricci-flat ALE metrics along the Ricci flow assuming an $L^{2}$-Łojasiewicz
inequality such as (0.2) holds true. This should be a quite general scheme of
proof and apply to other stability questions on _non-compact spaces_ along the
Ricci flow and other parabolic geometric flows.

######  Theorem 0.1 (Stability of Ricci-flat ALE metrics).

Let $(N^{n},g_{b})$ be a Ricci-flat ALE manifold of dimension $n\geqslant 4$
and $\frac{n-2}{2}<\tau<n-2$. There exists $\alpha\in(0,1)$ such that if we
assume:

  1. (1)

that $g_{b}$ is a local maximum of $\lambda_{\operatorname{ALE}}$ in the
$C^{2,\alpha}_{\tau}(g_{b})$ topology,

  2. (2)

in a $C^{2,\alpha}_{\tau}(g_{b})$-neighborhood
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$ of $g_{b}$, an
$L^{2}$-Łojasiewicz inequality is satisfied: for any metric $g$ in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$, we have

$|\lambda_{\operatorname{ALE}}(g)|^{2-\theta}\leq
C\|\nabla\lambda_{\operatorname{ALE}}(g)\|_{L^{2}(g_{b})}^{2}$

for some $\theta\in(0,1)$,

then, for any $0<\tau^{\prime}<\tau$ and $0<\alpha^{\prime}<\alpha$, for any
metric $g$ sufficiently $C^{2,\alpha}_{\tau}(g_{b})$-close to $g_{b}$, the
Ricci flow starting at $g$,
$C^{2,\alpha^{\prime}}_{\tau^{\prime}}(g_{b})$-converges to a Ricci-flat
metric $g^{\prime}_{b}$ (which is $C^{2,\alpha}_{\tau}(g_{b})$-close to
$g_{b}$) at a polynomial speed determined by the exponent $\theta$.

Theorem 0.1 provides Type IIb solutions of the Ricci flow unless the
background Ricci flat metric is the Euclidean metric, i.e. it ensures the
existence of immortal solutions $(g(t))_{t\geq 0}$ satisfying
$\limsup_{t\rightarrow+\infty}t\sup_{N}|\mathop{\rm
Rm}\nolimits(g(t))|=+\infty$. Notice also that Theorem 0.1 ensures the
convergence to hold in weighted Hölder spaces which we believe are well-suited
for gluing methods. We refer the reader to the article [BK17] for such an
illustration. Finally, we underline the need in Theorem 0.1 of restricting the
convergence rate in space below the threshold value $n-2$: we refer the reader
to our discussion on previous results of Dai and Ma [DM07] right after the
proof of Proposition 3.15 which links the mass and the mean value of the
scalar curvature along the solutions provided by Theorem 0.1.

A direct consequence of Theorem 1.12 and Theorem 0.1 is the following
stability result for integrable ALE Ricci-flat spaces of dimension at least
$5$.

######  Corollary 0.2.

Let $(N,g_{b})$ be a Ricci-flat ALE manifold of dimension $n\geqslant 5$ with
integrable Ricci-flat deformations. Assume that $g_{b}$ is _linearly stable_ ,
that is: its Lichnerowicz operator is nonpositive on divergence-free
deformations decaying at infinity. Let $\frac{n-2}{2}<\tau<n-2$ and
$0<\alpha<1$ sufficiently small.

Then, for any $0<\tau^{\prime}<\tau$ and $0<\alpha^{\prime}<\alpha$, for any
metric $g$ sufficiently $C^{2,\alpha}_{\tau}(g_{b})$-close to $g_{b}$, the
Ricci flow starting at $g$
$C^{2,\alpha^{\prime}}_{\tau^{\prime}}(g_{b})$-converges to a Ricci-flat
metric $g^{\prime}_{b}$ (which is $C^{2,\alpha}_{\tau}(g_{b})$-close to
$g_{b}$) at a polynomial speed determined by the dimension $n$.

Notice that Corollary 0.2 applies to all known Ricci-flat ALE metrics in
dimension greater than or equal to $5$. For instance, it applies to Calabi’s
ALE Ricci-flat Kähler metrics [Cal79] on the total space of the line bundle
$L^{-n}\rightarrow\mathbb{CP}^{n-1}$, $n\geq 3$, and more generally to Joyce’s
ALE Ricci-flat Kähler metrics [Joy00, Chapter $8$]. In particular, this makes
precise the results of [CT10] in the ALE case, where the convergence is
established on compact subsets only.

Corollary 0.2 shares some similarities with the work of Kröncke and Petersen
[KP20] that we now explain. In [KP20], the authors investigates the dynamical
stability of integrable ALE Ricci-flat metrics which carry a parallel spinor
in dimension greater than or equal to $4$. As they notice, this applies to all
know ALE Ricci-flat metrics. Their method is based on a delicate analysis of
the heat kernel of the Lichnerowicz operator in $L^{p}$ spaces and they apply
it to the DeTurck-Ricci flow with time-dependent background metrics. We
emphasize that the convergence result they get takes place in the $C^{k}\cap
L^{p}$ topology and the corresponding polynomial convergence rate is sharp
whereas Theorem 0.1 (and Corollary 0.2) proves the convergence of the Ricci
flow directly and in weighted spaces at the cost of getting an a priori non
sharp convergence rate. Moreover, they assume the initial condition to be a
perturbation of the ALE Ricci flat background to lie in $L^{p}\cap L^{\infty}$
for any $p<n$. In particular, they allow initial conditions to decay like
$r^{-1-\epsilon}$ at infinity while we require a decay rate to be at least
$r^{-(n-2)/2}$. Another interesting technical fact is that [KP20] considers
nearby Ricci-flat metrics of a given ALE Ricci flat metric with respect to the
Bianchi gauge whereas we study such metrics which are divergence free with
respect to a background ALE Ricci-flat metric. In the setting of Theorem 0.1,
we prove that the Bianchi gauge converges to $0$ faster then expected: see
Section 3.3 for a precise statement. Therefore, in view of the previous
remarks, we ask whether our methods can be carried to dimension $4$.

The next result echoes the work [HM14] on the existence of ancient solutions
coming out of an unstable Ricci-flat metric:

######  Theorem 0.3 (Instability of Ricci-flat ALE metrics).

Let $(N^{n},g_{b})$ be a Ricci-flat ALE manifold of dimension $n\geqslant 4$
and assume for some $0<\alpha<1$ and $\tau>\frac{n-2}{2}$:

  1. (1)

that $g_{b}$ is a _not_ a local maximum of $\lambda_{\operatorname{ALE}}$ in
the $C^{2,\alpha}_{\tau}(g_{b})$ topology,

  2. (2)

in a $C^{2,\alpha}_{\tau}(g_{b})$-neighborhood $\mathcal{U}_{g_{b}}$ of
$g_{b}$, an $L^{2}$-Łojasiewicz inequality is satisfied: for any metric $g$ in
$\mathcal{U}_{g_{b}}$, we have

$|\lambda_{\operatorname{ALE}}(g)|^{2-\theta}\leq
C\|\nabla\lambda_{\operatorname{ALE}}(g)\|_{L^{2}(g_{b})}^{2}$

for some $\theta\in(0,1)$.

Then, there exists a non Ricci-flat ancient solution to the Ricci flow
$(g(t))_{t\in(-\infty,0]}$ which is uniformly ALE of order $\tau$ and
$C^{2,\alpha}_{\tau}$-converges to $g_{b}$ at a polynomial speed determined by
the exponent $\theta$.

It is worth mentioning that it is still an open problem whether there are
unstable ALE Ricci flat metrics in dimension greater than or equal to $4$.
Also, Theorem 0.3 bears some resemblance with the work [Tak14] where an
ancient solution coming out of the Euclidean Schwarzschild metric is
constructed by hand. It would be an interesting problem to recover Takahashi’s
result via a suitable ALF version of our functional
$\lambda_{\operatorname{ALE}}$.

Both Theorem 0.1 and Theorem 0.3 rely heavily on Gaussian estimates and
estimates in weighted Hölder spaces for the heat kernel which we believe are
of independent interest and that we describe now.

### Heat kernel estimates

We prove that the heat kernel associated to the forward heat equation acting
on functions along a Ricci flow in a suitable neighborhood of a Ricci-flat ALE
metric satisfies uniform-in-time Gaussian bounds in Theorem 2.6. The proof
follows the method of Grigor’yan [Gri97] and that of Zhang [Zha06] in a Ricci
flow setting.

These controls are then used on the parabolic equation satisfied by the Ricci
curvature and integrated in time to yield controls on the metric in
$C^{2,\alpha}_{\tau}$ for $\frac{n-2}{2}<\tau<n-2$ as long as the Ricci flow
stays in a given $C^{2,\alpha}_{\tau}$-neighborhood of a Ricci-flat metric.
Indeed, lemma 2.12 starts with establishing short-time estimates for weighted
Hölder norm of the curvature operator and the Ricci tensor. Its proof already
reveals a constraint on the Hölder exponent $\alpha\in(0,1)$ in terms of the
weight $\tau$. Lemma 2.14 takes care of estimating the distance of a Ricci
flow satisfying mild assumptions from an ALE Ricci flat metric in the weighted
$C_{\tau}^{0}$ norm a priori. Then Lemma 2.15 proves an a priori bound for the
weighted $C^{0,\alpha}_{\tau+2}$ of the Ricci tensor. Finally, Lemma 2.17
establishes an a priori bound for the distance of a Ricci flow from an ALE
Ricci flat metric in the full weighted Hölder $C_{\tau+2}^{2,\alpha}$ norm in
terms of the weighted norm of the corresponding Ricci tensor. As expected, it
is more tractable to bound the Ricci tensor than the metric itself along such
a Ricci flow.

### Outline of paper

In Section 1, we give the main basic definitions of the article and review the
properties of the functional $\lambda_{\operatorname{ALE}}$ proven in [DO20].

In Section 2, we prove Gaussian bounds for the heat kernel along a Ricci flow
in a $C^{2,\alpha}_{\tau}$-neighborhood of a Ricci-flat ALE metric and deduce
controls on the flow in suitable weighted Hölder spaces.

In Section 3, using the previous controls on the flow thanks to the heat
kernel, we prove our stability result for metrics which are local maxima of
$\lambda_{\operatorname{ALE}}$ and around which a suitable $L^{2}$-Łojasiewicz
inequality holds.

Finally, in Section 4, we study the case when a metric is _not_ a local
maximum of the functional $\lambda_{\operatorname{ALE}}$ and prove that if a
suitable $L^{2}$-Łojasiewicz inequality holds around it, then, there exists an
ancient nontrivial Ricci flow coming out of it.

In the Appendix, Section A, we recall some formulas for the first and second
variations of the geometric quantities of interest here.

### Acknowledgements

We thank Klaus Kröncke for his suggestions and comments on a preliminary
version of this paper.

The first author is supported by grant ANR-17-CE40-0034 of the French National
Research Agency ANR (Project CCEM).

##  1\. The functional $\lambda_{\operatorname{ALE}}$ and its properties

In this section, we recall some of the properties of the functional
$\lambda_{\operatorname{ALE}}$ introduced and studied in [DO20]. This
functional and more precisely its sign in neighborhoods of Ricci-flat ALE
metrics (see Definition 1.1) will determine the dynamical stability or
instability of these Ricci-flat ALE metrics along the Ricci flow.

###  1.1. First definitions and function spaces

We start by defining the manifolds as well as the function spaces we will be
interested in.

######  Definition 1.1 (Asymptotically locally Euclidean (ALE) manifolds).

We will call a Riemannian manifold $(N,g)$ _asymptotically locally Euclidean_
(ALE) of order $\tau>0$ if the following holds: there exists a compact set
$K\subset N$, a radius $R>0$, $\Gamma$ a subgroup of $SO(n)$ acting freely on
$\mathbb{S}^{n-1}$ and a diffeomorphism
$\Phi:(\mathbb{R}^{n}/\Gamma)\backslash B_{e}(0,R)\mapsto N\backslash K$ such
that, if we denote $g_{e}$ the Euclidean metric on $\mathbb{R}^{n}/\Gamma$, we
have, for all $k\in\mathbb{N}$,

$\rho^{k}\big{|}\nabla^{g_{e},k}(\Phi^{*}g-g_{e})\big{|}_{e}=O(\rho^{-\tau}),$

on $\big{(}\mathbb{R}^{n}/\Gamma\big{)}\backslash B_{e}(0,R)$, where
$\rho=d_{e}(.,0)$.

We will study ALE metrics in a neighborhood of a Ricci flat ALE metric. Let us
start by defining this neighborhood thanks to weighted norms :

######  Definition 1.2 (Weighted Hölder norms for ALE spaces).

Let $(N,g,p)$ be an ALE manifold of dimension $n$, $\beta>0$, and
$\rho_{g}(x):=\max\big{(}d_{g}(x,p),1\big{)}$. For any tensor $s$, we define
the following weighted $C^{k,\alpha}_{\beta}$-norm :

$\|s\|^{g}_{C^{k,\alpha}_{\beta}(N)}:=\sup_{N}\rho_{g}^{\beta}\Big{(}\sum_{i=0}^{k}\rho_{g}^{i}|\nabla^{g,i}s|_{g}+\rho_{g}^{k+\alpha}[\nabla^{g,k}s]_{C^{\alpha}}\Big{)}.$

We will make a constant use of the following compact embedding whose proof can
be found for instance in [CSCB79, Lemme 3]:

######  Lemma 1.3.

Let $(N^{n},g)$ be an ALE manifold of dimension $n$. If $k\geq 0$, $\tau>0$
and $\alpha\in(0,1)$ then the continuous embeddings
$C^{k,\alpha}_{\tau}\hookrightarrow C^{k,\alpha^{\prime}}_{\tau^{\prime}}$ are
compact for $\alpha^{\prime}\in(0,\alpha)$ and $\tau^{\prime}\in(0,\tau)$.

The next definition concerns weighted Sobolev norms for ALE spaces:

######  Definition 1.4 (Weighted Sobolev norms for ALE spaces).

Let $\beta>0$, and $(N,g,p)$ an ALE manifold of dimension $n$, and
$\rho_{g}(x):=\max\big{(}d_{g}(x,p),1\big{)}$. For any tensor $s$, we define
the following weighted $L_{\beta}^{2}$-norm :

$\|s\|_{L^{2}_{\beta}}^{g}:=\Big{(}\int_{N}|s|^{2}\rho_{g}^{2\beta-n}dv_{g}\Big{)}^{\frac{1}{2}}.$

We moreover define the $H^{k}_{\beta}$-norm of $s$ as

$\|s\|_{H^{k}_{\beta}}^{g}:=\sum_{i=0}^{k}{\|\nabla^{i}s\|^{g}_{L^{2}_{\beta+i}}}.$

######  Remark 1.5.

Since the above definitions do not formally depend on the type of tensor, and
since the metrics we will consider will be equivalent, we will often abusively
simply denote these spaces $C^{k,\alpha}_{\tau}$ or $H^{k}_{\beta}$ for
instance.

###  1.2. The functionals $\lambda_{\operatorname{ALE}}^{0}$,
$m_{\operatorname{ADM}}$ and $\lambda_{\operatorname{ALE}}$

Next, as in [DM07, LP87, Bar86] let us consider for $\tau>\frac{n-2}{2}$ and
$\alpha\in(0,1)$, the following classical space of metrics,

$\mathcal{M}_{\tau}:=\left\\{\text{$g$ is a metric on $N$}\,|\,g-g_{b}\in
C^{1,\alpha}_{\tau}(S^{2}T^{*}N)\,|\,\mathop{\rm R}\nolimits_{g}\in
L^{1}\right\\},$ (1.1)

####  1.2.1. Mass of ALE metrics

On $\mathcal{M}_{\tau}$, the mass of an ALE metric is well-defined and only
depends on the metric:

$m_{\operatorname{ADM}}(g):=\lim_{R\rightarrow+\infty}\int_{\\{\rho_{g_{b}}=R\\}}\left<\mathop{\rm
div}\nolimits_{g_{b}}(g-g_{b})-\nabla^{g_{b}}\mathop{\rm
tr}\nolimits_{g_{b}}(g-g_{b}),\textbf{n}\right>_{g_{b}}\,d\sigma_{g_{b}},$
(1.2)

where n denotes the outward unit normal of the closed smooth hypersurfaces
$\\{\rho_{g_{b}}=R\\}$ for $R$ large.

######  Remark 1.6.

Given $g_{b}$ a Ricci-flat metric, it is clear from the definition that the
map $h\mapsto m_{\operatorname{ADM}}(g_{b}+h)$ is linear.

####  1.2.2. The functional $\lambda_{\operatorname{ALE}}^{0}$

######  Definition 1.7 ($\lambda_{\operatorname{ALE}}^{0}$, a first
renormalized Perelman’s functional).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric and let
$g\in\mathcal{M}^{2,\alpha}_{\tau}(g_{b},\varepsilon)$. Define the
$\mathcal{F}_{\operatorname{ALE}}$-energy by:

$\displaystyle\mathcal{F}_{\operatorname{ALE}}(w,g):=\int_{N}\big{(}4|\nabla^{g}w|_{g}^{2}+\mathop{\rm
R}\nolimits_{g}w^{2}\big{)}\,d\mu_{g},$ (1.3)

where $w-1\in C^{\infty}_{c}(N)$, where $C^{\infty}_{c}(N)$ is the space of
compactly supported smooth functions. The
$\lambda_{\operatorname{ALE}}^{0}$-functional associated to the
$\mathcal{F}_{\operatorname{ALE}}$-energy is:

$\lambda_{\operatorname{ALE}}^{0}(g):=\inf_{w}\mathcal{F}_{\operatorname{ALE}}(w,g),$

where the infimum is taken over functions $w:N\rightarrow\mathbb{R}$ such that
$w-1\in C_{c}^{\infty}(N)$.

We prove in [DO20] that the above infimum is attained by the (unique) solution
$w_{g}\in 1+C^{2,\alpha}_{\tau}$ to the equation

$-4\Delta_{g}w_{g}+\mathop{\rm R}\nolimits_{g}w_{g}=0.$ (1.4)

It turns out that $w_{g}$ is a positive function which lets us to consider the
potential function associated to a metric $g$ defined as

$f_{g}:=-2\ln w_{g},$

which is the unique solution $f_{g}\in C^{2,\alpha}_{\tau}$ to

$2\Delta_{g}f_{g}-|\nabla^{g}f_{g}|^{2}_{g}+\mathop{\rm R}\nolimits_{g}=0.$
(1.5)

####  1.2.3. Definition of the functional $\lambda_{\operatorname{ALE}}$

The above functionals $m_{\operatorname{ADM}}$ and
$\lambda_{\operatorname{ALE}}^{0}$ are only well-defined when the scalar
curvature is integrable which is not a convenient assumption as for instance
$\mathcal{M}_{\tau}$ is not a closed subset of $C^{2,\alpha}_{\tau}$ for
$\frac{n-2}{2}<\tau<n-2$. Moreover, $m_{\operatorname{ADM}}$ and
$\lambda_{\operatorname{ALE}}^{0}$ are not continuous with respect to the
$C^{2,\alpha}_{\tau}$-topology.

######  Remark 1.8.

Another finer topology for $\mathcal{M}_{\tau}$ is obtained by adding the
$L^{1}$-norm of the scalar curvature. We will see in the rest of the present
article that there are Ricci flows of ALE metric with nonvanishing (or even
infinite) mass $C^{2,\alpha}_{\tau}$-converging to a Ricci-flat ALE metric.
This implies that the scalar curvature does not converge in an $L^{1}$-sense
by [DM07, Corollary 3].

In order to solve these problems, we define for $g$ in a small
$C^{2,\alpha}_{\tau}$-neighborhood of a Ricci-flat ALE metric $g_{b}$ and in
$\mathcal{M}_{\tau}$, the functional

$\lambda_{\operatorname{ALE}}(g):=\lambda_{\operatorname{ALE}}^{0}(g)-m_{\operatorname{ADM}}(g).$

The advantage is that $\lambda_{\operatorname{ALE}}$ extends as an analytic
function on a $C^{2,\alpha}_{\tau}$-neighborhood of any Ricci-flat ALE metric,
see [DO20, Proposition 3.4]. Moreover, by denoting $w_{g}$ the solution to
(1.4), which is well-defined without assuming that the scalar curvature is
integrable, we have

$\begin{split}\lambda_{\operatorname{ALE}}(g)=\lim_{R\rightarrow+\infty}\Bigg{(}\int_{\\{\rho_{g_{b}}\leq
R\\}}&\left(|\nabla^{g}f_{g}|^{2}_{g}+\mathop{\rm
R}\nolimits_{g}\right)\,e^{-f_{g}}d\mu_{g}\\\
&-\int_{\\{\rho_{g_{b}}=R\\}}\left<\mathop{\rm
div}\nolimits_{g_{b}}(g)-\nabla^{g_{b}}\mathop{\rm
tr}\nolimits_{g_{b}}(g),\textbf{n}_{g_{b}}\right>_{g_{b}}\,d\sigma_{g_{b}}\Bigg{)}.\end{split}$
(1.6)

It is worth noting from [DO20, Example 3.1] that for most perturbations, none
of the above integrals converge as $R\to\infty$, but their difference always
does if $g$ is sufficiently $C^{2,\alpha}_{\tau}$-close to $g_{b}$ for
$\tau>\frac{n}{2}-1$.

###  1.3. Main properties of $\lambda_{\operatorname{ALE}}$

We now list some of the properties of $\lambda_{\operatorname{ALE}}$ proven in
[DO20].

####  1.3.1. Variations of $\lambda_{\operatorname{ALE}}$

We have the following first and second variations for
$\lambda_{\operatorname{ALE}}$.

######  Proposition 1.9 (First variation of $\lambda_{\operatorname{ALE}}$).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric asymptotic to
$\mathbb{R}^{n}/\Gamma$, for some finite subgroup $\Gamma$ of $SO(n)$ acting
freely on $\mathbb{S}^{n-1}$, and let $\tau\in(\frac{n-2}{2},n-2)$.

The first variation of $\lambda_{\operatorname{ALE}}$ on a neighborhood of
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ at $g$ in the direction $h$ is:

$\begin{split}\delta_{g}\lambda_{\operatorname{ALE}}(h)=&-\int_{N}\langle
h,\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\rangle_{g}\,e^{-f_{g}}d\mu_{g}.\end{split}$
(1.7)

and the tensor $\mathop{\rm Ric}\nolimits(g)+\nabla^{g,2}f_{g}$ is weighted
divergence free, i.e.

$\mathop{\rm div}\nolimits_{f_{g}}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)=0,$ (1.8)

where $\mathop{\rm div}\nolimits_{f_{g}}T=\mathop{\rm
div}\nolimits_{g}T-T(\nabla^{g}f_{g})$ for a symmetric $2$-tensor $T$.

Moreover, if $(g(t))_{t\in[0,T)}$ is a solution to the Ricci flow on $N$ lying
in $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$, then we have the following
monotonicity formula,

$\frac{d}{dt}\lambda_{\operatorname{ALE}}(g(t))=2\|\mathop{\rm
Ric}\nolimits(g(t))+\nabla^{g(t),2}f_{g(t)}\|_{L^{2}(e^{-f_{g(t)}}d\mu_{g(t)})}^{2}\geq
0.$ (1.9)

We next consider the second variations of $\lambda_{\operatorname{ALE}}$.

######  Definition 1.10.

Let $(N,g)$ be a Riemmanian metric. Then the _Lichnerowicz operator_
associated to $g$ acting on symmetric $2$-tensors, denoted by $L_{g}$, is
defined by:

$L_{g}h:=\Delta_{g}h+2\mathop{\rm Rm}\nolimits(g)(h)-\mathop{\rm
Ric}\nolimits(g)\circ h-h\circ\mathop{\rm Ric}\nolimits(g),\quad h\in
C_{loc}^{2}(S^{2}T^{*}N),$ (1.10)

where $\Delta_{g}=-\nabla_{g}^{*}\nabla_{g}$ and where $\mathop{\rm
Rm}\nolimits(g)(h)(X,Y):=h(\mathop{\rm Rm}\nolimits(g)(e_{i},X)Y,e_{i})$ for
an orthonormal basis $(e_{i})_{i=1}^{n}$ with respect to $g$. In particular,
if $(N^{n},g_{b})$ is an ALE Ricci flat metric, then,

$L_{g}h:=\Delta_{g}h+2\mathop{\rm Rm}\nolimits(g)(h),\quad h\in
C_{loc}^{2}(S^{2}T^{*}N).$ (1.11)

######  Proposition 1.11 (Second variation of $\lambda_{\operatorname{ALE}}$
at a Ricci flat metric).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric asymptotic to
$\mathbb{R}^{n}/\Gamma$, for some finite subgroup $\Gamma$ of $SO(n)$ acting
freely on $\mathbb{S}^{n-1}$, and let $\tau\in(\frac{n-2}{2},n-2)$. Then the
second variation of $\lambda_{\operatorname{ALE}}$ at $g_{b}$ along a
divergence free variation $h\in S^{2}T^{*}N$ such that $h\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ is:

$\delta^{2}_{g_{b}}\lambda_{\operatorname{ALE}}(h,h)=\frac{1}{2}\langle
L_{g_{b}}h,h\rangle_{L^{2}}.$ (1.12)

####  1.3.2. A Łojasiewicz inequality

As explained in the Introduction of this paper, in [DO20], we moreover proved
that in a $C^{2,\alpha}_{\tau}$-neighborhood of any Ricci-flat ALE space, a
Łojasiewicz inequality holds for $\lambda_{\operatorname{ALE}}$.

######  Theorem 1.12 ([DO20]).

Let $(N^{n},g_{b})$ be an ALE Ricci-flat manifold of dimension $n\geq 4$. Let
$\alpha\in(0,1)$ and $\tau\in(\frac{n-2}{2},n-2)$. Then there exist a
neighborhood $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ of $g_{b}$, a
constant $C>0$ and $\theta\in(0,1]$ such that for any metric $g\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$, we have the following
$L^{2}_{\frac{n}{2}+1}$-Łojasiewicz inequality,

$|\lambda_{\operatorname{ALE}}(g)|^{2-\theta}\leq
C\|\nabla\lambda_{\operatorname{ALE}}(g)\|_{L^{2}_{\frac{n}{2}+1}(g_{b})}^{2}.$
(1.13)

Moreover, if $(N^{n},g_{b})$ has integrable infinitesimal Ricci-flat
deformations, then $\theta=1$.

In particular, if $n\geq 5$, one has the following $L^{2}$-Łojasiewicz
inequality for integrable Ricci-flat ALE metrics: if
$\tau\in(\frac{n}{2},n-2)$ then for any
$0<\delta<\frac{2\tau-(n-2)}{2\tau-(n-4)}$, there exists $C>0$ such that for
all $g\in B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$,

$|\lambda_{\operatorname{ALE}}(g)|^{2-\theta_{L^{2}}}\leq
C\|\nabla\lambda_{\operatorname{ALE}}(g)\|_{L^{2}(g_{b})}^{2},\quad\theta_{L^{2}}:=2-\frac{1}{\delta}=\frac{2\tau-n}{2\tau-(n-2)}.$
(1.14)

Here $\nabla\lambda_{\operatorname{ALE}}$ denotes the gradient of
$\lambda_{\operatorname{ALE}}$ in the $L^{2}(g)$ sense.

##  2\. Preliminary estimates for short and large time

###  2.1. Heat kernel Gaussian bounds

In this section, we fix an ALE Ricci flat metric $(N^{n},g_{b})$ once and for
all. Let $(g(t))_{t\in[0,T)}$ be a solution to the Ricci flow with $g(0)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ with $\tau\in(\frac{n-2}{2},n-2)$.
The main aim of this section is to establish Gaussian bounds for the heat
kernel acting on functions along the Ricci flow.

As explained in the Introduction, we follow the same strategy adopted by
Grigor’yan [Gri97] and Zhang [Zha06].

We denote the heat kernel associated to the forward heat equation by
$K(x,t,y,s)$, for $0\leq s<t<T$, and $x,y\in N$:

$\left\\{\begin{array}[]{rl}&\partial_{t}K(\cdot,\cdot,y,s)=\Delta_{g(t)}K(\cdot,\cdot,y,s),\\\
&\\\ &\partial_{t}g=-2\mathop{\rm Ric}\nolimits(g(t)),\\\ &\\\
&\lim_{t\rightarrow s^{+}}K(\cdot,t,y,s)=\delta_{y},\end{array}\right.$ (2.1)

on $N\times(0,T)$. This heat kernel always exists and is positive: see
[Gue02].

We also consider its conjugate heat equation: if $(x,t)\in N\times(0,T)$ is
fixed, then $K(x,t,\cdot,\cdot)$ is the heat kernel associated to the
conjugate backward heat equation:

$\left\\{\begin{array}[]{rl}&\partial_{s}K(x,t,\cdot,\cdot)=-\Delta_{g(s)}K(x,t,\cdot,\cdot)+\mathop{\rm
R}\nolimits_{g(s)}K(x,t,\cdot,\cdot),\\\ &\\\ &\partial_{s}g=-2\mathop{\rm
Ric}\nolimits(g(s)),\\\ &\\\ &\lim_{s\rightarrow
t^{-}}K(x,t,\cdot,s)=\delta_{x}.\end{array}\right.$ (2.2)

on $N\times(0,t)$.

We start with the following proposition that estimates the $L^{1}$ norms of
both forward and backward heat kernels:

######  Proposition 2.1 ($L^{1}$-bound).

With the setting and notations introduced above, if $0\leq s\leq t<T$,

$\displaystyle e^{-\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}}$ $\displaystyle\leq$
$\displaystyle\int_{N}K(x,t,y,s)\,d\mu_{g(t)}(x)\leq
e^{\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}},\quad y\in N,$ (2.3)
$\displaystyle 1$ $\displaystyle=$
$\displaystyle\int_{N}K(x,t,y,s)\,d\mu_{g(s)}(y),\quad\quad x\in N.$ (2.4)

###### Proof.

Let $(\Omega_{j})_{j\geq 0}$ be an increasing sequence of domains of $N$ with
smooth boundary exhausting the manifold $N$. Let $K_{j}(\cdot,\cdot,y,s)$ be
the heat kernel associated to (2.1) on $\Omega_{j}$ with Dirichlet boundary
condition. Then, one can prove that $(K_{j}(\cdot,\cdot,y,s))_{j\geq 0}$ is an
increasing sequence converging locally uniformly to $K(\cdot,\cdot,y,s)$. By
integrating by parts, one gets, for some fixed $(y,s)\in N\times[0,T)$,

$\displaystyle\partial_{t}\int_{\Omega_{j}}K_{j}(x,t,y,s)d\mu_{g(t)}(x)$
$\displaystyle=$
$\displaystyle\int_{\Omega_{j}}\Delta_{g(t)}K_{j}(x,t,y,s)-\mathop{\rm
R}\nolimits_{g(t)}(x)K_{j}(x,t,y,s)\,d\mu_{g(t)}(x)$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega_{j}}\left<\nabla^{g(t)}_{x}K_{j}(x,t,y,s),\mathbf{n}\right>\,d\sigma_{j,g(t)}(x)$
$\displaystyle-\int_{\Omega_{j}}\mathop{\rm
R}\nolimits_{g(t)}(x)K_{j}(x,t,y,s)\,d\mu_{g(t)}(x)$ $\displaystyle\leq$
$\displaystyle-\int_{\Omega_{j}}\mathop{\rm
R}\nolimits_{g(t)}(x)K_{j}(x,t,y,s)\,d\mu_{g(t)}(x)$ $\displaystyle\leq$
$\displaystyle\|\mathop{\rm
R}\nolimits_{g(t)}\|_{C^{0}}\int_{\Omega_{j}}K_{j}(x,t,y,s)\,d\mu_{g(t)}(x),$

where $d\sigma_{j,g(t)}$ is the induced measure on $\partial\Omega_{j}$ by
$d\mu_{g(t)}$. Therefore, by Grönwall’s inequality:

$\begin{split}\int_{\Omega_{j}}K_{j}(x,t,y,s)\,d\mu_{g(t)}(x)&\leq
e^{\int_{t^{\prime}}^{t}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}ds}\int_{\Omega_{j}}K_{j}(x,t^{\prime},y,s)\,d\mu_{g(t^{\prime})}(x),\\\
\end{split}$

for any $t\geq t^{\prime}>s$. Let $j$ large enough so that $y\in\Omega_{j}$
and let $t^{\prime}$ go to $s$ to get:

$\begin{split}\int_{\Omega_{j}}K_{j}(x,t,y,s)\,d\mu_{g(t)}(x)&\leq
e^{\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}},\end{split}$

for any $t>s$. By letting $j$ tending to $+\infty$, we get a half of the first
estimate (2.3), i.e.

$\begin{split}\int_{N}K(x,t,y,s)\,d\mu_{g(t)}(x)&\leq
e^{\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}}.\end{split}$ (2.5)

On the other hand, let $(\phi_{k})_{k}$ be any sequence of smooth cut-off
functions approximating the constant function $1$ with values into $[0,1]$
such that
$\lim_{k\rightarrow+\infty}\|\nabla^{g(t^{\prime}),k}\phi_{k}\|_{C^{0}}=0$ for
$k=1,2$ and uniformly in $t^{\prime}\in[s,t]$. Then

$\begin{split}\partial_{t}\int_{N}\phi_{k}(x)K(x,t,y,s)\,d\mu_{g(t)}(x)&=-\int_{N}\Delta_{g(t)}\phi_{k}(x)K(x,t,y,s)\,d\mu_{g(t)}(x)\\\
&-\int_{N}\phi_{k}(x)\mathop{\rm
R}\nolimits_{g(t)}(x)K(x,t,y,s)\,d\mu_{g(t)}(x)\\\
&\geq-e^{\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}}\sup_{\mathop{\rm
supp}\nolimits(\phi_{k})}\arrowvert\Delta_{g(t)}\phi_{k}\arrowvert\\\
&-\|R_{g(t)}\|_{C^{0}}\int_{N}\phi_{k}(x)K(x,t,y,s)\,d\mu_{g(t)}(x).\end{split}$

Here, we have used (2.5) in the second inequality. Again, by Grönwall’s
inequality:

$\begin{split}\int_{N}\phi_{k}(x)K(x,t,y,s)\,d\mu_{g(t)}(x)\geq
e^{-\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}}\left(1-C\int_{s}^{t}\sup_{\mathop{\rm
supp}\nolimits(\phi_{k})}\arrowvert\Delta_{g(t^{\prime})}\phi_{k}\arrowvert\,dt^{\prime}\right),\end{split}$

which implies the expected estimate by letting $k$ go to $+\infty$.

The second estimate (2.4) is proved similarly. Remark that
$K_{j}(x,t,\cdot,\cdot)$ satisfies (2.2) with Dirichlet boundary condition.
Therefore,

$\displaystyle\int_{\Omega_{j}}K_{j}(x,t,y,s)\,d\mu_{g(s)}(y)\leq 1,\quad
s<t,$

for any index $j$ large enough. This implies

$\displaystyle\int_{N}K(x,t,y,s)\,d\mu_{g(s)}(y)\leq 1,\quad s<t.$

Similarly, one can prove the expected lower bound for
$\int_{N}K(x,t,y,s)\,d\mu_{g(s)}(y)$.

∎

######  Remark 2.2.

In particular, Proposition 2.1 tells us that the heat semigroup associated to
$\Delta_{g(t)}$ is a bounded operator when interpreted as an operator on
$L^{1}$ and $L^{\infty}$. By interpolation, one gets for any
$p\in[1,+\infty]$,

$\displaystyle\|u_{t}\|_{L^{p}(d\mu_{g(t)})}\leq
e^{\frac{1}{p}\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}}\|u_{s}\|_{L^{p}(d\mu_{g(s)})},\quad
t>s,$ $\displaystyle u_{t}(x):=K(x,t,\cdot,s)\ast u_{s}.$

The next proposition concerns an on diagonal upper bound for the forward heat
kernel along the Ricci flow.

######  Proposition 2.3 (On diagonal upper bound: $L^{2}\rightarrow
L^{\infty}$ bound).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let $(g(t))_{t\in[0,T)}$ be a
solution to the Ricci flow such that $g(t)\in B_{C^{0}}(g_{b},\varepsilon)$
for all $t\in[0,T)$ and such that

$\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2}.$

Then,

$0<K(x,t,y,s)\leq\frac{C}{(t-s)^{\frac{n}{2}}},\quad 0\leq s<t,\quad x,y\in
N,$

where $C=C(n,g_{b},\varepsilon)$ is a time-independent positive constant.

######  Remark 2.4.

A more accurate but also more ad-hoc assumption on the smallness of the time
integral of the $C^{0}$ norm of the scalar curvature would be

$\int_{\frac{s+t}{2}}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2}.$

We start with the following lemma that is crucial to prove Proposition 2.3:

######  Lemma 2.5 ($L^{1}$ mean value inequality).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let $(g(t))_{t\in[0,T)}$ be a
solution to the Ricci flow such that $g(t)\in B_{C^{0}}(g_{b},\varepsilon)$
for all $t\in[0,T)$. Then any nonnegative subsolution $u$ of the heat equation
along such a Ricci flow, i.e.

$\partial_{t}u\leq\Delta_{g(t)}u,\quad\mbox{on $N\times(0,T)$,}$

satisfies, for $\eta\in(0,1)$ and $r^{2}<t$ such that
$\int_{t-r^{2}}^{t}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds\leq\frac{1}{2}$,

$\displaystyle\sup_{P\left(x,t,\eta r\right)}u\leq
C\fint_{P(x,t,r)}u\,d\mu_{g(s)}ds,$

for some positive constant $C=C(n,g_{b},\varepsilon,\eta)$ and where

$P(x,t,r):=\left\\{(y,s)\in N\times[0,T)\quad|\quad s\in(t-r^{2},t],\quad y\in
B_{g_{b}}(x,r)\right\\}.$

The proof of Lemma 2.5 is inspired by that of Zhang [Zha06] in a Ricci flow
setting. This was already noticed in [DL17] in the context of expanding
gradient Ricci solitons. Therefore, we only sketch a proof of Lemma 2.5 for
the convenience of the reader.

###### Proof of Lemma 2.5.

Let $p\in[1,+\infty)$. Then the function $u^{p}$ is a sub-solution to the heat
equation, i.e.

$\partial_{t}u^{p}\leq\Delta_{g(t)}u^{p},$ (2.6)

on $N\times(0,T)$. Consider any smooth space-time cutoff function $\psi$ and
multiply (2.6) by $\psi^{2}u^{p}$ and integrate by parts as follows:

$\begin{split}&\int_{t-r^{2}}^{t^{\prime}}\int_{N}\arrowvert\nabla^{g(s)}(\psi
u^{p})\arrowvert^{2}_{g(s)}-\arrowvert\nabla^{g(s)}\psi\arrowvert^{2}_{g(s)}u^{2p}\,d\mu_{g(s)}ds\leq\\\
&\int_{t-r^{2}}^{t^{\prime}}\int_{N}\frac{\partial_{s}(\psi^{2}\ln
d\mu_{g(s)})}{2}u^{2p}\,d\mu_{g(s)}ds-\frac{1}{2}\int_{N}\psi^{2}u^{2p}\,d\mu_{g(t^{\prime})},\end{split}$

for any $t^{\prime}\in(t-r^{2},t].$

Hence,

$\begin{split}&\int_{t-r^{2}}^{t^{\prime}}\int_{N}\arrowvert\nabla^{g(s)}(\psi
u^{p})\arrowvert^{2}_{g(s)}\,d\mu_{g(s)}ds+\frac{1}{2}\int_{N}\psi^{2}u^{2p}\,d\mu_{g(t^{\prime})}\leq\\\
&\int_{t-r^{2}}^{t^{\prime}}\int_{N}\left(\frac{\partial_{s}\psi^{2}}{2}+\arrowvert\nabla^{g(s)}\psi\arrowvert_{g(s)}^{2}+\frac{1}{2}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\psi^{2}\right)u^{2p}\,d\mu_{g(s)}ds\leq\\\
&\int_{t-r^{2}}^{t^{\prime}}\int_{N}\left(\frac{\partial_{s}\psi^{2}}{2}+\arrowvert\nabla^{g(s)}\psi\arrowvert_{g(s)}^{2}\right)u^{2p}\,d\mu_{g(s)}ds\\\
&+\frac{1}{2}\int_{t-r^{2}}^{t^{\prime}}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds\sup_{s\in(t-r^{2},t^{\prime}]}\int_{N}\psi^{2}u^{2p}\,d\mu_{g(s)}\leq\\\
&\int_{t-r^{2}}^{t^{\prime}}\int_{N}\left(\frac{\partial_{s}\psi^{2}}{2}+\arrowvert\nabla^{g(s)}\psi\arrowvert_{g(s)}^{2}\right)u^{2p}\,d\mu_{g(s)}ds+\frac{1}{2}\int_{t-r^{2}}^{t}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds\sup_{s\in(t-r^{2},t^{\prime}]}\int_{N}\psi^{2}u^{2p}\,d\mu_{g(s)}.\end{split}$
(2.7)

Notice in particular that (LABEL:inequ-IBP-nash-moser-pre) implies, if
$\int_{t-r^{2}}^{t}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds\leq\frac{1}{2}$,

$\sup_{s\in(t-r^{2},t^{\prime}]}\int_{N}\psi^{2}u^{2p}\,d\mu_{g(s)}\leq
4\int_{t-r^{2}}^{t^{\prime}}\int_{N}\left(\frac{\partial_{s}\psi^{2}}{2}+\arrowvert\nabla^{g(s)}\psi\arrowvert_{g(s)}^{2}\right)u^{2p}\,d\mu_{g(s)}ds.$
(2.8)

Let $\tau,\sigma\in(0,+\infty)$ such that $\tau\geq\sigma$ and
$\tau+\sigma<r$. In particular, $P(x,t,\tau)\subset P(x,t,\tau+\sigma)\subset
P(x,t,r)$. Now, choose two smooth functions
$\phi:\mathbb{R}_{+}\rightarrow[0,1]$ and
$\eta:\mathbb{R}_{+}\rightarrow[0,1]$ such that

$\displaystyle\mathop{\rm
supp}\nolimits(\phi)\subset[0,\tau+\sigma],\quad\phi\equiv 1\quad\mbox{in
$[0,\tau]$},\quad\phi\equiv 0\quad\mbox{in
$[\tau+\sigma,+\infty)$},\quad-c/\sigma\leq\phi^{\prime}\leq 0,$
$\displaystyle\mathop{\rm
supp}\nolimits(\eta)\subset[t-(\tau+\sigma)^{2},+\infty),\quad\eta\equiv
1\quad\mbox{in $[t-\tau^{2},+\infty)$},$ $\displaystyle\eta\equiv
0\quad\mbox{in $(t-r^{2},t-(\tau+\sigma)^{2}]$},\quad 0\leq\eta^{\prime}\leq
c/\sigma^{2}.$

Define $\psi(y,s):=\phi(d_{g_{b}}(x,y))\eta(s)$, for $(y,s)\in N\times(0,T)$.
Then,

$\displaystyle\arrowvert\nabla^{g(s)}\psi\arrowvert_{g(s)}\leq\frac{C}{\sigma},\quad\arrowvert\partial_{s}\psi\arrowvert\leq\frac{C}{\sigma^{2}},$

for some time-independent positive constant $C$. Here we have used the fact
that $g(t)$ is $\varepsilon$ $C^{0}$-close to $g_{b}$. Now,
$(N^{n},g(t))_{t\in(0,T)}$ satisfies the following Euclidean Sobolev
inequality since $(N^{n},g_{b})$ does by [Chap.$3$, [SC02]] and the fact that
$g(t)$ is $\varepsilon$ $C^{0}$-close to $g_{b}$: there exists $C=C(g_{b})>0$
such that

$\left(\int_{N}(\psi
u^{p})^{\frac{2n}{n-2}}\,d\mu_{g(t)}\right)^{\frac{n-2}{n}}\leq
C\int_{N}\arrowvert\nabla^{g(t)}(\psi
u^{p})\arrowvert_{g(t)}^{2}\,d\mu_{g(t)},$ (2.9)

for any $t\in[0,T)$. We are now in a position to follow [DL17] very closely:
if $\alpha_{n}:=1+2/n$, one gets by Hölder’s inequality together with (2.9),

$\begin{split}\int_{P(x,t,\tau)}\left(u^{2p}\right)^{\alpha_{n}}\,d\mu_{g(s)}ds\leq&\int_{P(x,t,r)}\left(\psi
u^{p}\right)^{2\alpha_{n}}\,d\mu_{g(s)}ds\\\
\leq&\int_{t-r^{2}}^{t}\left(\int_{N}(\psi
u^{p})^{\frac{2n}{n-2}}\,d\mu_{g(s)}\right)^{\frac{n-2}{n}}\left(\int_{N}(\psi
u^{p})^{2}\,d\mu_{g(s)}\right)^{\frac{2}{n}}ds\\\
\leq&C\sup_{s\in(t-r^{2},t]}\left(\int_{N}(\psi
u^{p})^{2}\,d\mu_{g(s)}\right)^{\frac{2}{n}}\int_{P(x,t,r)}\arrowvert\nabla^{g(s)}(\psi
u^{p})\arrowvert_{g(s)}^{2}\,d\mu_{g(s)}\\\
\leq&C\frac{1}{\sigma^{2\alpha_{n}}}\left(\int_{P(x,t,\tau+\sigma)}u^{2p}\,d\mu_{g(s)}\right)^{\alpha_{n}},\end{split}$

for some time-independent positive constant $C$. Here, we have used
(LABEL:inequ-IBP-nash-moser-pre) and (2.8) in the last line together with the
fact that $\psi$ is compactly supported in $P(x,t,\tau+\sigma)$. It is now
sufficient to iterate the previous inequality for suitable sequences
$(p_{i})_{i}$, $(r_{i})_{i}$ and $(\tau_{i})_{i}$, $(\sigma_{i})_{i}$ to reach
the desired conclusion.

∎

We are in a position to give a proof of Proposition 2.3.

###### Proof of Proposition 2.3.

It suffices to apply Lemma 2.5 together with Proposition 2.1 to the
nonnegative (sub)solution

$u(x,t):=K(x,t,y,s),$

for some fixed $(y,s)\in N\times[0,T)$ with $r^{2}=(t-s)/2$:

$\begin{split}K(x,t,y,s)\leq&\sup_{P\left(x,t,\frac{1}{2}\sqrt{\frac{t-s}{2}}\right)}K(\cdot,\cdot,y,s)\\\
\leq&\frac{C}{(t-s)^{\frac{n+2}{2}}}\int_{P(x,t,\sqrt{\frac{t-s}{2}})}K(x^{\prime},t^{\prime},y,s)\,d\mu_{g(t^{\prime})}(x^{\prime})dt^{\prime}\\\
\leq&\frac{C}{(t-s)^{\frac{n}{2}}}e^{\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}},\end{split}$

for some positive constant $C=C(n,g_{b},\varepsilon).$ This ends the proof of
the expected estimate since $\int_{s}^{t}\|\mathop{\rm
R}\nolimits_{g(t^{\prime})}\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2}$.

∎

We now state the main result of this section:

######  Theorem 2.6 (Gaussian estimate).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let $(g(t))_{t\in[0,T)}$ be a
solution to the Ricci flow such that $g(t)\in B_{C^{0}}(g_{b},\varepsilon)$
for all $0\leqslant s\leqslant t<T$ and such that

$\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2n}.$ (2.10)

Then the heat kernel associated to (2.1) satisfies the following Gaussian
estimate:

$\begin{split}&|K(x,t,y,s)|\leq\frac{C}{(t-s)^{\frac{n}{2}}}\exp\left\\{-\frac{d_{g(s)}^{2}(x,y)}{D(t-s)}\right\\},\quad
0\leq s<t,\quad x,y\in N,\quad D>D_{0},\end{split}$ (2.11)

where $C=C(n,g_{b},\varepsilon,D)$ and $D_{0}=D_{0}(n,g_{b},\varepsilon)$ are
time-independent positive constants.

######  Remark 2.7.

The Gaussian weight in (2.11) could be stated in terms of the distance
$d_{g(t)}(x,y)$ instead thanks to Proposition A.2.

###### Proof.

Again, we follow closely the presentation of [DL17]. Define the following
integral quantities for a positive constant $D$ to be chosen later:

$\begin{split}&E^{1}_{D}(s,t,z):=\int_{N}|K(x,t,z,s)|^{2}e^{\frac{2d_{g(t)}^{2}(x,z)}{D(t-s)}}\,d\mu_{g(t)}(x),\quad
s<t,\quad z\in N,\\\
&E^{2}_{D}(s,t,z):=\int_{N}|K(z,t,y,s)|^{2}e^{\frac{2d_{g(s)}^{2}(z,y)}{D(t-s)}}\,d\mu_{g(s)}(y),\quad
s<t,\quad z\in N.\\\ \end{split}$

The motivation for introducing such quantities follows from the next crucial
observation: by the semi-group property and the triangular inequality,

$\begin{split}K(x,t,y,s)&=\int_{N}K(x,t,z,\tau)\cdot
K(z,\tau,y,s)\,d\mu_{g(\tau)}(z)\\\
&\leq\int_{N}K(x,t,z,\tau)e^{\frac{d_{g(\tau)}^{2}(x,z)}{D(t-\tau)}}K(z,\tau,y,s)e^{\frac{d_{g(\tau)}^{2}(z,y)}{D(\tau-s)}}\,d\mu_{g(\tau)}(z)e^{-\frac{d_{g(\tau)}^{2}(x,y)}{D(t-s)}}\\\
\end{split}$

where $s<\tau:=\frac{t+s}{2}<t.$ Therefore, by Cauchy-Schwarz inequality, we
get the following "universal" inequality:

$|K(x,t,y,s)|\leq
C\sqrt{E^{1}_{D}(s,\tau,y)E^{2}_{D}(\tau,t,x)}\exp\left\\{-\frac{d_{g(\tau)}^{2}(x,y)}{D(t-s)}\right\\}.$
(2.12)

We claim the following:

######  Claim 1.

$E^{1}_{D}(s,t,\cdot)+E^{2}_{D}(s,t,\cdot)\leq\frac{C}{(t-s)^{n/2}},\quad
s<t,\quad D>D_{0},$ (2.13)

for some time-independent positive constants $C=C(n,g_{b},\varepsilon,D)$ and
$D_{0}=D_{0}(n,g_{b},\varepsilon)$.

The proof of Claim 1 is essentially based on Proposition 2.3 implied by (2.10)
and is virtually identical to the proof given in [DL17]: it will therefore be
omitted.

Now, thanks to (2.12) together with Claim 1, one obtains the expected Gaussian
estimate:

$\displaystyle|K(x,t,y,s)|$ $\displaystyle\leq$
$\displaystyle\frac{C}{(t-s)^{\frac{n}{2}}}\exp\left\\{-\frac{d_{g(\tau)}^{2}(x,y)}{D(t-s)}\right\\}$
$\displaystyle\leq$
$\displaystyle\frac{C}{(t-s)^{\frac{n}{2}}}\exp\left\\{-\frac{d_{g(s)}^{2}(x,y)}{e^{\frac{1}{n}}\cdot
D(t-s)}\right\\},\quad 0\leq s<t,\quad x,y\in N,$

for some positive constant $C=C(n,g_{b},\varepsilon,D)$ and any $D\geq
D_{0}(n,g_{b},\varepsilon)$. Here we have used Proposition A.2 to estimate
from below the distance $d_{g(\tau)}(x,y)$ in terms of the distance
$d_{g(s)}(x,y)$.

∎

###  2.2. $L^{2}-C^{0}$ estimate

We start by checking that the $L^{2}$-norms of the Ricci tensor and the
Hessian of the potential function satisfying (1.5) are controlled from above
by that of the associated Bakry-Émery tensor.

######  Proposition 2.8 (A priori $L^{2}$ estimate for the Ricci flow).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let $g$ be a metric in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$. Then,

$\begin{split}\|\mathop{\rm
Ric}\nolimits(g)\|_{L^{2}}+\|\nabla^{g,2}f_{g}\|_{L^{2}}\leq
C\left(\|\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\|_{L^{2}}\right),\end{split}$ (2.14)

for some positive constant $C=C(n,g_{b},\varepsilon)$.

###### Proof.

Let $g$ be a metric in $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$. Then
observe that from the Euler-Lagrange equation (1.5) satisfied by the potential
function $f_{g}$, one gets:

$\Delta_{f_{g}}f_{g}=-\Delta_{g}f_{g}-\mathop{\rm R}\nolimits_{g}=-\mathop{\rm
tr}\nolimits_{g}\left(\mathop{\rm Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right).$
(2.15)

Let us apply the Bochner formula to the smooth metric measure space
$\left(N^{n},g,\nabla^{g}f_{g}\right)$:

$\begin{split}\Delta_{f_{g}}|\nabla^{g}f_{g}|^{2}_{g}=&2|\nabla^{g,2}f_{g}|^{2}_{g}+2\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)(\nabla^{g}f_{g},\nabla^{g}f_{g})\\\
&+2\langle\nabla^{g}f_{g},\nabla^{g}\Delta_{f_{g}}f_{g}\rangle_{g}\\\
=&2|\nabla^{g,2}f_{g}|^{2}_{g}+2\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)(\nabla^{g}f_{g},\nabla^{g}f_{g})\\\
&-2\left\langle\nabla^{g}f_{g},\nabla^{g}\mathop{\rm
tr}\nolimits_{g}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\right\rangle_{g}.\end{split}$

Here, we have used (2.15) in the second line. By integrating by parts (with
respect to the weighted measure $e^{-f_{g}}d\mu_{g}$) the previous identity,
one ends up with:

$\begin{split}2\|\nabla^{g,2}f_{g}\|^{2}_{L^{2}}=&-2\left\langle\Delta_{f_{g}}f_{g},\mathop{\rm
tr}\nolimits_{g}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\right\rangle_{L^{2}}\\\
&-2\int_{N}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)(\nabla^{g}f_{g},\nabla^{g}f_{g})\,e^{-f_{g}}d\mu_{g}\\\
=&2\left\|\mathop{\rm tr}\nolimits_{g}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\right\|^{2}_{L^{2}}\\\
&-2\int_{N}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)(\nabla^{g}f_{g},\nabla^{g}f_{g})\,e^{-f_{g}}d\mu_{g}\\\
\leq&2\left\|\mathop{\rm tr}\nolimits_{g}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\right\|^{2}_{L^{2}}\\\
&+\|\rho_{g_{b}}^{2}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\|_{C^{0}}\|\rho_{g_{b}}^{-1}\nabla^{g}f_{g}\|_{L^{2}}^{2}\\\
\leq&2\left\|\mathop{\rm tr}\nolimits_{g}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\right\|^{2}_{L^{2}}\\\
&+C(n,g_{b})\|\rho_{g_{b}}^{2}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\|_{C^{0}}\|\nabla^{g,2}f_{g}\|^{2}_{L^{2}}.\end{split}$
(2.16)

Here we have used (2.15) in the second line, and Hardy’s inequality from
[Min09] is invoked in the last inequality. In particular, if $g$ lies in a
sufficiently $C^{2,\alpha}_{\tau}$ small neighborhood of $g_{b}$ such that
$C(n,g_{b})\|\rho_{g_{b}}^{2}(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g})\|\leq 1$ then the last term on the
righthand side of (2.16) can be absorbed by the lefthand side so that,

$\|\nabla^{g,2}f_{g}\|^{2}_{L^{2}}\leq 2\left\|\mathop{\rm
tr}\nolimits_{g}\left(\mathop{\rm
Ric}\nolimits(g)+\nabla^{g,2}f_{g}\right)\right\|^{2}_{L^{2}}.$ (2.17)

This ends the proof of the estimate on the Hessian term in (2.14). The bound
on the $L^{2}$-norm of the Ricci curvature follows by the triangular
inequality. ∎

The next proposition lets us get an a priori $C^{0}$ estimate on the distance
from a Ricci flow to the origin given by an ALE Ricci flat metric:

######  Proposition 2.9 (A priori $L^{2}-C^{0}$ estimate for the Ricci flow).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let $(g(t))_{t\in[0,T)}$ be a
solution to the Ricci flow such that $g(t)\in B_{C^{0}}(g_{b},\varepsilon)$
and with bounded curvature for all $t\in[0,T)$.

For $t\in(0,T)$ and $0<r^{2}<t$, if $\int_{t-r^{2}}^{t}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds\leq\frac{1}{2}$ and $\mathop{\rm
Ric}\nolimits(g(s))\in L^{2}$, $s\in[t-r^{2},t]$, then,

$\displaystyle\|\mathop{\rm Ric}\nolimits(g(t))\|_{C^{0}}$ $\displaystyle\leq$
$\displaystyle Cr^{-\frac{n}{2}}\exp\left(c(n)\int_{t-r^{2}}^{t}\|\mathop{\rm
Rm}\nolimits(g(s))\|_{C^{0}}\,ds\right)\|\mathop{\rm
Ric}\nolimits(g(t-r^{2})\|_{L^{2}},$ (2.18)

for some positive constant $C=C(n,g_{b},\varepsilon)$.

###### Proof.

Thanks to the evolution equation [(A.6), Lemma A.3] satisfied by the Ricci
tensor, one gets that the function
$u(t):=\exp\left(-c(n)\int_{0}^{t}\|\mathop{\rm
Rm}\nolimits(g(s))\|_{C^{0}}\,ds\right)|\mathop{\rm
Ric}\nolimits(g(t)|_{g(t)}$ is a subsolution to the heat equation, i.e.
$\partial_{t}u\leq\Delta_{g(t)}u$ in the weak sense.

Thanks to Lemma 2.5, if $r^{2}<t\leq T$ and $x\in N$, one is led to:

$\begin{split}&\sup_{B_{g_{b}}(x,\frac{r}{2})}|\mathop{\rm
Ric}\nolimits(g(t))|^{2}_{g(t)}\leq\frac{C}{r^{n+2}}e^{c(n)\int_{t-r^{2}}^{t}\|\mathop{\rm
Rm}\nolimits(g(s))\|_{C^{0}}\,ds}\int_{t-r^{2}}^{t}\int_{B_{g_{b}}(x,r)}|\mathop{\rm
Ric}\nolimits(g(s))|^{2}\,d\mu_{g(s)}ds,\end{split}$ (2.19)

where $C=C(n,g_{b},\varepsilon)$ is a positive constant which is independent
of space and time variables.

This being said, estimate (2.19) implies in particular that

$\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0}}\leq\frac{C(n,g_{b},\varepsilon)}{r^{\frac{n}{2}}}e^{c(n)\int_{t-r^{2}}^{t}\|\mathop{\rm
Rm}\nolimits(g(s))\|_{C^{0}}\,ds}\sup_{t-r^{2}\leq s\leq t}\|\mathop{\rm
Ric}\nolimits(g(s))\|_{L^{2}}.$ (2.20)

Now, Proposition A.6 gives:

$\sup_{t-r^{2}\leq s\leq t}\|\mathop{\rm Ric}\nolimits(g(s))\|_{L^{2}}\leq
e^{c(n)\int_{t-r^{2}}^{t}\|\mathop{\rm
Rm}\nolimits(g(s))\|_{C^{0}}\,ds}\|\mathop{\rm
Ric}\nolimits(g(t-r^{2}))\|_{L^{2}}.$ (2.21)

Combining this fact with (2.20) finally give

$\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0}}\leq\frac{C(n,g_{b},\varepsilon)}{r^{\frac{n}{2}}}e^{c(n)\int_{t-r^{2}}^{t}\|\mathop{\rm
Rm}\nolimits(g(s))\|_{C^{0}}\,ds}\|\mathop{\rm
Ric}\nolimits(g(t-r^{2}))\|_{L^{2}}.$ (2.22)

Then (2.22) establishes the desired estimate (2.18). ∎

We end this section by an estimate on the decay of the $C^{0}$-norm of the
Ricci tensor along a Ricci flow satisfying some mild assumptions:

######  Proposition 2.10.

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let $(g(t))_{t\in[0,T)}$ be a
solution to the Ricci flow such that $g(t)\in B_{C^{0}}(g_{b},\varepsilon)$
and with bounded curvature for all $t\in[0,T)$ and such that

$\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2n}.$ (2.23)

Then, if $p\geq 1$, $0<s<t<T$ and $\mathop{\rm Ric}\nolimits(g(s))\in L^{p}$,

$\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0}}\leq\frac{C}{(t-s)^{\frac{n}{2p}}}\|\mathop{\rm
Ric}\nolimits(g(s))\|_{L^{p}}+C\sup_{t^{\prime}\in[0,T)}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0}}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime},$ (2.24)

for some time-independent positive constant $C=C(n,g_{b},\varepsilon,p).$

######  Remark 2.11.

Proposition 2.10 will be most useful in case $p>\frac{n}{\tau+2}$ since we
will ultimately consider a solution $(g(t))_{t\in[0,T)}$ to the Ricci flow
such that $\mathop{\rm Ric}\nolimits(g(t))=O(\rho_{g_{b}}^{-\tau-2})$ for each
time $t\in[0,T)$.

###### Proof.

The assumptions legitimate the use of Theorem 2.6 through Duhamel’s formula as
we now explain.

According to the evolution equation satisfied by the Ricci curvature along the
Ricci flow given by Lemma (A.3) together with Duhamel’s formula, one can
write:

$\begin{split}\mathop{\rm
Ric}\nolimits(g(t))(x)=&\int_{N}\left<\mathcal{K}(x,t,\cdot,s),\mathop{\rm
Ric}\nolimits(g(s))\right>\,d\mu_{g(s)}\\\
&+\int_{s}^{t}\int_{N}\left<\mathcal{K}(x,t,\cdot,t^{\prime}),\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\right>\,d\mu_{g(t^{\prime})}\,dt^{\prime},\end{split}$
(2.25)

where $\mathcal{K}(x,t,y,s)$ denotes the heat kernel acting on symmetric
$2$-tensors associated to the one-parameter family of metrics
$(g(t))_{t\in[0,T)}$.

Now, by Kato’s inequality, the operator norm of $\mathcal{K}(x,t,y,s)$ is
bounded by the heat kernel acting on functions, i.e.

$\|\mathcal{K}(x,t,y,s)\|_{Hom(S^{2}T_{y}^{*}N,S^{2}T_{x}^{*}N)}\leq
K(x,t,y,s),\quad(x,y)\in N\times N,\,\,t>s\geq 0,$

where $K(x,t,y,s)$ denotes the heat kernel acting on functions with respect to
the metric $g(t)$ introduced and studied in Section 2.1. In particular,
[(2.11), Theorem 2.6] ensures (once we fix $D=D_{0}$) that

$\|K(x,t,\cdot,s)\|_{L^{q}}\leq\frac{C(n,g_{b})}{(t-s)^{\frac{n}{2}\left(1-\frac{1}{q}\right)}},$

for $0<s<t<T.$ Therefore, one gets by Hölder’s inequality and (2.25):

$\begin{split}\|\mathop{\rm Ric}\nolimits(g(t))\|_{C^{0}}\leq\,&\sup_{x\in
N}\|K(x,t,\cdot,s)\|_{L^{q}}\|\mathop{\rm Ric}\nolimits(g(s))\|_{L^{p}}\\\
&+\sup_{t^{\prime}\in[0,T)}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0}}\int_{s}^{t}\|K(x,t,\cdot,s)\|_{L^{1}}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\\\
\leq\,&\frac{C(n,g_{b})}{(t-s)^{\frac{n}{2p}}}\|\mathop{\rm
Ric}\nolimits(g(s))\|_{L^{p}}+C\sup_{t^{\prime}\in[0,T)}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0}}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime},\end{split}$

where $p,q\geq 1$ are such that $p^{-1}+q^{-1}=1$ and where $C$ is a positive
constant independent of time.

∎

###  2.3. Weighted estimates

In this section, we consider an ALE Ricci flat metric $(N^{n},g_{b})$ and
prove an a priori $C^{0}$ estimate on the distance of a Ricci flow
$(g(t))_{t\in[0,T)}$ lying in a small neighborhood
$B_{C^{0}}(g_{b},\varepsilon)$ starting from a metric $g(0)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$.

To do so, we essentially use the Gaussian estimates proved in Theorem 2.6 via
the Duhamel’s formula. We will distinguish short-time from large-time
estimates.

######  Lemma 2.12 (Short-time estimates).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let
$\tau\in(\frac{n-2}{2},n-2)$, $\alpha\in\left(0,\min\\{1,n-2-\tau\\}\right)$
and let $(g(t))_{t\in[0,T)}$ be a solution to the Ricci flow in a neighborhood
$B_{C^{0}}(g_{b},\varepsilon)$ such that $g(0)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ and

$\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2n}.$ (2.26)

Then for $t\in[0,T)\,\cap\,[0,T_{\operatorname{Shi}}]$,

$\displaystyle\|\mathop{\rm Rm}\nolimits(g(t))\|_{C^{0,\alpha}_{\tau+2}}\leq
C(n,g_{b},\varepsilon)e^{C(n,g_{b},\varepsilon)\cdot t}\|\mathop{\rm
Rm}\nolimits(g(0))\|_{C^{0,\alpha}_{\tau+2}},$ (2.27)
$\displaystyle\|\mathop{\rm Ric}\nolimits(g(t))\|_{C^{0,\alpha}_{\tau+2}}\leq
C(n,g_{b},\varepsilon)e^{C(n,g_{b},\varepsilon)\cdot t}\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0,\alpha}_{\tau+2}}.$ (2.28)

######  Remark 2.13.

The proof of Lemma 2.12 uses Theorem 2.6 which explains the assumption (2.26).
Using the maximum principle and suitable barriers in the same spirit as it is
done in [Li18] would have led to the same result without assuming (2.26).

###### Proof.

According to (A.14) from Proposition A.5, the curvature operator is uniformly
bounded on $[0,T)\,\cap\,[0,T_{\operatorname{Shi}}]$. Thanks to the evolution
equation satisfied by the curvature operator [(A.5), Lemma A.3], this implies
that the function $e^{-c(n)\|\mathop{\rm Rm}\nolimits(g(0))\|_{C^{0}}\cdot
t}|\mathop{\rm Rm}\nolimits(g(t))|_{g(t)}$ is a subsolution to the heat
equation along the Ricci flow. In particular, by Duhamel’s formula together
with the maximum principle,

$\begin{split}|\mathop{\rm
Rm}\nolimits(g(t))|_{g(t)}(x)\leq\,&e^{c(n)\|\mathop{\rm
Rm}\nolimits(g(0))\|_{C^{0}}\cdot t}\int_{N}K(x,t,y,0)|\mathop{\rm
Rm}\nolimits(g(0))|_{g(0)}(y)\,d\mu_{g(0)}(y)\\\ \leq\,&e^{C\cdot
t}\int_{N}K(x,t,y,0)|\mathop{\rm
Rm}\nolimits(g(0))|_{g(0)}(y)\,d\mu_{g(0)}(y),\end{split}$

for some positive constant $C=C(n,g_{b},\varepsilon)$ by assumption on $g(0)$.
By assumption, [(2.11), Theorem 2.6] holds true and we get (once we fix
$D=D_{0}$),

$\begin{split}|\mathop{\rm
Rm}\nolimits(g(t))|_{g(t)}(x)&\leq\frac{C}{t^{\frac{n}{2}}}\int_{N}\exp\left\\{-\frac{d_{g(0)}^{2}(x,y)}{D_{0}t}\right\\}|\mathop{\rm
Rm}\nolimits(g(0))|_{g(0)}(y)\,d\mu_{g(0)}(y)\\\ &\leq C\|\mathop{\rm
Rm}\nolimits(g(0))\|_{C^{0}_{\tau+2}}\int_{N}t^{-\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(x,y)}{2D_{0}t}\right\\}\rho_{g_{b}}^{-\tau-2}(y)\,d\mu_{g_{b}}(y),\end{split}$
(2.29)

where we have used $d_{g(0)}^{2}(x,y)\geq\frac{1}{2}d_{g_{b}}^{2}(x,y)$ for
all $x,y\in N$ in the last line. By splitting the integral on the righthand
side of (2.29) in two parts, whether $y\in B_{g_{b}}(p,d_{g_{b}}(p,x)/2)$ or
not for some fixed point $p\in N$, one gets:

$\begin{split}\int_{N}t^{-\frac{n}{2}}&\exp\left\\{-\frac{d_{g_{b}}^{2}(x,y)}{2D_{0}t}\right\\}\rho_{g_{b}}^{-\tau-2}(y)\,d\mu_{g_{b}}(y)\leq\\\
&\quad
t^{-\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(p,x)}{8D_{0}t}\right\\}\int_{B_{g_{b}}(p,d_{g_{b}}(p,x)/2)}\rho_{g_{b}}^{-\tau-2}(y)\,d\mu_{g_{b}}(y)\\\
&\quad+\frac{C}{(d_{g_{b}}(p,x)+1)^{\tau+2}}\int_{B^{c}_{g_{b}}(p,d_{g_{b}}(p,x)/2)}t^{-\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(x,y)}{2D_{0}t}\right\\}\,d\mu_{g_{b}}(y)\\\
&\leq
C\left(\frac{d_{g_{b}}^{2}(p,x)}{t}\right)^{\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(p,x)}{8D_{0}t}\right\\}(d_{g_{b}}(p,x)+1)^{-\tau-2}+\frac{C}{(d_{g_{b}}(p,x)+1)^{\tau+2}}\\\
&\leq\frac{C}{(d_{g_{b}}(p,x)+1)^{\tau+2}},\end{split}$ (2.30)

for some positive constant $C$ independent of time $t>0$ that may vary from
line to line. Here, we have used the fact that $\tau+2<n$ in the second line.
Estimates (2.29) and (2.30) lead to the expected result (2.27) at the level of
the $C^{0}$-norm:

$\|\mathop{\rm Rm}\nolimits(g(t))\|_{C^{0}_{\tau+2}}\leq Ce^{C\cdot
t}\|\mathop{\rm Rm}\nolimits(g(0))\|_{C^{0}_{\tau+2}},$ (2.31)

for some positive constant $C=C(n,g_{b},\varepsilon).$

Regarding the $C^{\alpha}$-seminorm, by a similar reasoning considering
difference quotients together with Duhamel’s formula , one gets schematically
if $\alpha\in\left(0,\min\\{1,n-2-\tau\\}\right)$:

$\begin{split}\rho_{g_{b}}^{\tau+2+\alpha}[\mathop{\rm
Rm}\nolimits(g(t))]_{C^{\alpha}}\leq\,&Ce^{C\cdot
t}\|\rho_{g_{b}}^{\tau+2+\alpha}[\mathop{\rm
Rm}\nolimits(g(0))]_{C^{\alpha}}\|_{C^{0}}\\\ &+C\int_{0}^{t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast[\mathop{\rm
Rm}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\|_{C^{0}_{\tau+2+\alpha}}\,dt^{\prime}\\\
\leq\,&Ce^{C\cdot t}\|\mathop{\rm
Rm}\nolimits(g(0))\|_{C^{0,\alpha}_{\tau+2}}\\\ &+Ce^{C\cdot t}\|\mathop{\rm
Rm}\nolimits(g(0))\|_{C^{0}}\int_{0}^{t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}_{\tau+2}}\,dt^{\prime}.\end{split}$
(2.32)

Concatenating (2.32) together with (2.31) leads to (2.27) by invoking
Grönwall’s inequality.

The proof of (2.28) goes along the same lines by using (A.6) together with
(2.27). ∎

The next lemma takes care of a priori integral-in-time estimates for the
weighted $C^{0}_{\tau}$-norm of the Ricci curvature.

######  Lemma 2.14 (A priori large-time $C^{0}$ estimate).

Let $(N^{n},g_{b})$ be an ALE Ricci flat metric. Let
$\tau\in(\frac{n-2}{2},n-2)$ and let $(g(t))_{t\in[0,T)}$ be a solution to the
Ricci flow in a neighborhood $B_{C^{0}}(g_{b},\varepsilon)$ such that
$\sup_{t\in[0,T)}\|\mathop{\rm Rm}\nolimits(g(t))\|_{C^{0}_{\tau+2}}\leq
C(n,g_{b})$ and

$\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2n}.$ (2.33)

Then we have

$\begin{split}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}&\leq
C\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))\|_{C^{0}}\,ds^{\prime}\right),\quad 0\leq
s<t<T,\end{split}$ (2.34)

for some time-independent positive constant $C=C(n,g_{b},\varepsilon)$. In
particular, one gets the following a priori $C^{0}$ estimate:

$\|g(t)-g(s)\|_{C^{0}_{\tau}}\leq C\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))\|_{C^{0}}\,ds^{\prime}\right),\quad 0\leq s<t<T.$
(2.35)

###### Proof.

By arguing as in the proof of Proposition 2.10, one ensures with the help of
Duhamel’s formula that for $x\in N$ and $t\geq s\geq 0$,

$\begin{split}|\mathop{\rm
Ric}\nolimits(g(t))|_{g(t)}(x)\leq&\int_{N}K(x,t,\cdot,s)|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}\,d\mu_{g(s)}\\\
&+\int_{s}^{t}\int_{N}K(x,t,\cdot,t^{\prime})|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast\mathop{\rm
Ric}\nolimits(g(t^{\prime}))|_{g(t^{\prime})}\,d\mu_{g(t^{\prime})}\,dt^{\prime}.\end{split}$
(2.36)

By invoking [(2.11), Theorem 2.6] (once we fix $D=D_{0}$), the first term on
the righthand side of (2.36) can be estimated as follows:

$\begin{split}K(x,t,\cdot,s)\ast|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}&:=\int_{N}K(x,t,\cdot,s)|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}\,d\mu_{g(s)}\\\
&\leq\frac{C}{(t-s)^{\frac{n}{2}}}\int_{N}\exp\left\\{-\frac{d_{g(s)}^{2}(x,y)}{D_{0}(t-s)}\right\\}|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}(y)\,d\mu_{g(s)}(y)\\\ &\leq C\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}\int_{N}(t-s)^{-\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(x,y)}{2D_{0}(t-s)}\right\\}\rho_{g_{b}}^{-\tau-2}(y)\,d\mu_{g_{b}}(y),\end{split}$
(2.37)

where we have used $d_{g(s)}^{2}(x,y)\geq\frac{1}{2}d_{g_{b}}^{2}(x,y)$ for
all $x,y\in N$ in the last line together with the fact that $g(t)\in
B_{C^{0}}(g_{b},\varepsilon)$ for all $t\in[0,T)$.

Observe that for $x\neq y$, and $n\geq 3$,

$\begin{split}\int_{s}^{+\infty}(t-s)^{-\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(x,y)}{2D_{0}(t-s)}\right\\}\,dt\leq
Cd_{g_{b}}^{-n+2}(x,y).\end{split}$

In particular, by Fubini’s theorem,

$\begin{split}\int_{s}^{t}K(x,t^{\prime},\cdot,s)\ast|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}\,dt^{\prime}\leq C\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}\int_{N}d_{g_{b}}^{-n+2}(x,y)\rho_{g_{b}}^{-\tau-2}(y)\,d\mu_{g_{b}}(y).\end{split}$

By considering regions of the type $B_{g_{b}}(p,d_{g_{b}}(p,x)/2)$,
$B_{g_{b}}(p,2d_{g_{b}}(p,x))\setminus B_{g_{b}}(p,d_{g_{b}}(p,x)/2)$ and
$N\setminus B_{g_{b}}(p,2d_{g_{b}}(p,x))$ for a point $x\in N$ and a fixed
point $p\in N$, one gets an a priori estimate on the following
$C^{0}_{\tau}$-norm:

$\int_{s}^{t}\|K(x,t^{\prime},\cdot,s)\ast|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}\|_{C^{0}_{\tau}}\,dt^{\prime}\leq C\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}.$ (2.38)

Notice that we heavily use the restriction $\tau<n-2$ here. We proceed
similarly to handle the second term on the righthand side of (2.36):

$\begin{split}\int_{s}^{t}&\left|\int_{s}^{t^{\prime}}\int_{N}\left<\mathcal{K}(x,t^{\prime},\cdot,s^{\prime}),\mathop{\rm
Rm}\nolimits(g(s^{\prime}))\ast\mathop{\rm
Ric}\nolimits(g(s^{\prime}))\right>\,d\mu_{g(s^{\prime})}\,ds^{\prime}\right|dt^{\prime}\leq\\\
&C\int_{s}^{t}\int_{N}\left(\int_{s^{\prime}}^{t}K(x,t^{\prime},y,s^{\prime})\,dt^{\prime}\right)|\mathop{\rm
Rm}\nolimits(g(s^{\prime}))|(y)\,d\mu_{g(s^{\prime})}(y)\|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))\|_{C^{0}}\,ds^{\prime}\\\ &\leq
C\left(\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))\|_{C^{0}}\,ds^{\prime}\right)\int_{N}d_{g_{b}}(x,y)^{-n+2}\rho_{g_{b}}(y)^{-\tau-2}\,d\mu_{g_{b}}(y)\\\
&\leq C\left(\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))\|_{C^{0}}\,ds^{\prime}\right)\rho_{g_{b}}(x)^{-\tau}.\end{split}$
(2.39)

Here, we have used Fubini’s theorem in the second line together with the
assumption on the curvature tensor $\mathop{\rm Rm}\nolimits(g(s^{\prime}))$,
$s^{\prime}\in[0,T)$ in the penultimate line which implies that the curvature
tensor $\mathop{\rm Rm}\nolimits(g(s^{\prime}))$ decays as fast as
$\rho_{g_{b}}^{-\tau-2}$ uniformly in time.

Estimates (2.38) and (2.39) lead directly to the desired estimate (2.34).

Finally, (2.35) is proved with the help of (2.34) by noticing that:

$\begin{split}\|g(t)-g(s)\|_{C^{0}_{\tau}}&\leq\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}.\end{split}$

∎

The following lemma proves an a priori weighted $C^{0,\alpha}$ estimate on the
Ricci tensor of a solution of the Ricci flow $C^{2,\alpha}_{\tau}$-close to a
stable ALE Ricci flat metric.

######  Lemma 2.15 (A priori weighted $C^{0,\alpha}$ estimate on Ricci
curvature).

Let $(N^{n},g_{b})$, $n\geq 4$, be an ALE Ricci flat metric. Let
$\tau\in(\frac{n-2}{2},n-2)$, $\alpha\in(0,1)$ and let $(g(t))_{t\in[0,T)}$ be
a solution to the Ricci flow in a neighborhood
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ with uniformly bounded covariant
derivatives, i.e. $\|\nabla^{g(t),k}\mathop{\rm
Rm}\nolimits(g(t))\|_{C^{0}}\leq C_{k}$, $k\geq 1$ and such that

$\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2n}.$ (2.40)

Then we have the following estimate,

$\begin{split}\|\mathop{\rm Ric}\nolimits(g(t))\|_{C^{0}_{\tau+2}}&\leq
C\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\right),\quad 0\leq
s<t<T,\end{split}$ (2.41)

for some time-independent positive constant $C=C(n,g_{b},\varepsilon)$.

Moreover, if $\alpha\in\left(0,\min\\{1,n-2-\tau\\}\right)$ then for any
$\eta\in(0,1)$:

$\begin{split}\|\mathop{\rm Ric}\nolimits(g(t))\|_{C^{0,\alpha}_{\tau+2}}&\leq
C_{\eta}\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0,\alpha}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{1-\eta}_{C^{0}}\,dt^{\prime}\right),\quad
0\leq s<t<T.\end{split}$ (2.42)

###### Proof.

Similarly to the beginning of the proof of Lemma 2.14, we use Duhamel’s
formula to estimate the $C^{0}_{\tau+2}$ norm of the Ricci curvature along
such a Ricci flow.

Using the Gaussian bounds (2.11) from Theorem 2.6 with $D=D_{0}$ fixed once
and for all, leads us to:

$\begin{split}&\left|\int_{N}\left<\mathcal{K}(x,t,\cdot,s),\mathop{\rm
Ric}\nolimits(g(s))\right>\,d\mu_{g(s)}\right|_{g(t)}\\\
&\leq\frac{C}{(t-s)^{\frac{n}{2}}}\int_{N}\exp\left\\{-\frac{d_{g(0)}^{2}(x,y)}{D_{0}(t-s)}\right\\}|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}(y)\,d\mu_{g(s)}(y)\\\ &\leq C\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}\int_{N}(t-s)^{-\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(x,y)}{2D_{0}(t-s)}\right\\}\rho_{g_{b}}^{-\tau-2}(y)\,d\mu_{g_{b}}(y).\end{split}$
(2.43)

Here we have used the fact that $g_{b}$ and $g(0)$ are uniformly equivalent
(due to their $\varepsilon$ closeness). By splitting the integral on the
righthand side of (LABEL:ric-est-c0-duha) in two parts, whether $y\in
B_{g_{b}}(p,d_{g_{b}}(p,x)/2)$ or not for some fixed point $p\in N$, one gets:

$\int_{N}(t-s)^{-\frac{n}{2}}\exp\left\\{-\frac{d_{g_{b}}^{2}(x,y)}{2D_{0}(t-s)}\right\\}\rho_{g_{b}}^{-\tau-2}(y)\,d\mu_{g_{b}}(y)\leq\frac{C}{(d_{g_{b}}(p,x)+1)^{\tau+2}},$
(2.44)

for some positive constant time-independent $C$ that may vary from line to
line. Here, we have used the fact that $\tau+2<n$ in the second line.
Concatenating estimates (LABEL:ric-est-c0-duha) and (2.44) give the following
estimates in terms of the initial data:

$\begin{split}\left\|\int_{N}\left<\mathcal{K}(x,t,\cdot,s),\mathop{\rm
Ric}\nolimits(g(s))\right>\,d\mu_{g(s)}\right\|_{C^{0}_{\tau+2}}\leq
C\|\mathop{\rm Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}},\quad t>s\geq
0.\end{split}$ (2.45)

We proceed similarly to estimate the second term on the right hand side of
(2.36):

$\begin{split}&\left\|\int_{s}^{t}\int_{N}\left<\mathcal{K}(x,t,\cdot,t^{\prime}),\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\right>\,d\mu_{g(t^{\prime})}\,dt^{\prime}\right\|_{C^{0}_{\tau+2}}\\\
&\leq C\int_{s}^{t}\|\mathop{\rm Rm}\nolimits(g(t^{\prime}))\ast\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau+2}}\,ds\\\ &\leq
C\sup_{t^{\prime}\in[s,t]}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0}_{2}}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}\\\ &\leq
C\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\right).\end{split}$ (2.46)

Here, we have used the fact that $g(t)$ is close to $g_{b}$ in the
$C^{2}_{\tau}$-topology in the penultimate line which implies in particular
that the curvature tensor $\mathop{\rm Rm}\nolimits(g(s))$ decays as fast as
$\rho_{g_{b}}^{-2}$ uniformly in time. Finally, [(2.34), Lemma 2.15] is
invoked in the last line. Notice that we have not used the whole strength of
the decay of the curvature tensor.

The proof of the a priori estimate on the $C^{0,\alpha}_{\tau+2}$ norm of the
Ricci tensor goes along the same lines as the $C^{0}_{\tau+2}$ a priori
estimate we just proved by using difference quotients under the restriction on
$\alpha\in(0,1)$ so that $\tau+2+\alpha<n.$

Indeed, under this restriction on $\alpha$, we get schematically:

$\begin{split}\rho_{g_{b}}^{\tau+2+\alpha}[\mathop{\rm
Ric}\nolimits(g(t))]_{C^{\alpha}}\leq\,&C\|\rho_{g_{b}}^{\tau+2+\alpha}[\mathop{\rm
Ric}\nolimits(g(s))]_{C^{\alpha}}\|_{C^{0}}+C\int_{s}^{t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast[\mathop{\rm
Ric}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\|_{C^{0}_{\tau+2+\alpha}}\,dt^{\prime}\\\
&+C\int_{s}^{t}\|[\mathop{\rm
Rm}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\ast\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau+2+\alpha}}\,dt^{\prime}\\\
\leq\,&C\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0,\alpha}_{\tau+2}}+C\int_{s}^{t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast[\mathop{\rm
Ric}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\|_{C^{0}_{\tau+2+\alpha}}\,dt^{\prime}\\\
&+C\sup_{t^{\prime}\in[s,t]}\|\rho_{g_{b}}^{\alpha}[\mathop{\rm
Rm}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\|_{C^{0}_{2}}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}\\\
\leq\,&C\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0,\alpha}_{\tau+2}}+C\int_{s}^{t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast[\mathop{\rm
Ric}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\|_{C^{0}_{\tau+2+\alpha}}\,dt^{\prime}\\\
&+C\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\right)\\\
\leq\,&C\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0,\alpha}_{\tau+2}}+C\int_{s}^{t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\ast[\mathop{\rm
Ric}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\|_{C^{0}_{\tau+2+\alpha}}\,dt^{\prime}\\\
&+C\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}.\end{split}$ (2.47)

Here we have used Lemma 2.14 in the third inequality.

In order to handle the last integral on the righthand side of (2.47), we
proceed as follows: by interpolation together with Hölder inequality,

$\begin{split}\int_{s}^{t}\rho_{g_{b}}(x)^{\alpha}&[\mathop{\rm
Ric}\nolimits(g(t^{\prime}))]_{C^{\alpha}}(x)\,dt^{\prime}\leq
C_{\alpha}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}^{1-\alpha}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{\alpha}_{C^{0}_{1}}\,dt^{\prime}\\\
\leq\,&C_{\alpha}\left(\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\right)^{1-\alpha}\cdot\left(\int_{s}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{1}}\,dt^{\prime}\right)^{\alpha}\\\
\end{split}$ (2.48)

for some time-independent positive constant $C_{\alpha}$. Now, by
interpolation inequalities from Lemma A.8, we have for any $k\geq 2$:

$\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{1}}\leq C_{k}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\frac{k}{k-1}}}^{1-\frac{1}{k}},$

since we assume all the covariant derivatives of the curvature tensor to be
bounded uniformly in time.

Therefore, by Hölder’s inequality, if $k\geq 2$ is sufficiently large so that
$1-\frac{1}{\tau}>\frac{1}{k}$,

$\begin{split}\int_{s}^{t}\|\nabla^{g(t)}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{1}}\,dt^{\prime}\leq\,&C_{k}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\frac{k}{k-1}}}^{1-\frac{1}{k}}\,dt^{\prime}\\\
\leq\,&C_{k}\left(\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}\right)^{\frac{1}{\tau}}\left(\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}^{1-\frac{\tau}{k(\tau-1)}}\,dt^{\prime}\right)^{1-\frac{1}{\tau}}.\end{split}$
(2.49)

Now, by Lemma 2.14 together with (2.48) and (2.49), one gets:

$\begin{split}\int_{s}^{t}\rho_{g_{b}}(x)^{\alpha}[\mathop{\rm
Ric}\nolimits(g(t^{\prime}))]_{C^{\alpha}}(x)\,dt^{\prime}\leq\,&C_{k,\alpha}\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\right)\\\
&+C_{k,\alpha}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}^{1-\frac{\tau}{k(\tau-1)}}\,dt^{\prime},\end{split}$
(2.50)

which implies the expected estimate (2.42) and ends the proof. ∎

######  Remark 2.16.

The proof of Lemma 2.15 uses the Gaussian estimates established in Section 2.1
in an essential way. Indeed, by linearizing the evolution equation satisfied
by the Ricci curvature along the Ricci flow at the metric $g_{b}$, given by
Lemma (A.3), one gets schematically:

$\begin{split}\partial_{t}\mathop{\rm
Ric}\nolimits(g(t))=\Delta_{g_{b}}\mathop{\rm
Ric}\nolimits(g(t))+Q(h(t),\mathop{\rm Ric}\nolimits(g(t))),\end{split}$
(2.51)

where $Q(h(t),\mathop{\rm Ric}\nolimits(g(t)))$ is a symmetric $2$-tensor on
$N$ such that pointwise,

$\begin{split}&\left|Q(h(t),\mathop{\rm Ric}\nolimits(g(t)))\right|\leq\\\
&C(n,g_{b},\varepsilon)\left(|\mathop{\rm
Rm}\nolimits(g_{b})|_{g_{b}}|\mathop{\rm
Ric}\nolimits(g(t))|_{g_{b}}+\sum_{k=0}^{2}\left|\nabla^{g_{b},k}(g(t)-g_{b})\right|_{g_{b}}\left|\nabla^{g_{b},2-k}\mathop{\rm
Ric}\nolimits(g(t))\right|_{g_{b}}\right).\end{split}$

Observe that $g_{b}$ has non-negative Ricci curvature (since it is Ricci flat)
and non-collapsed at all scales, i.e. $\mathop{\rm
Vol}\nolimits_{g_{b}}B_{g_{b}}(x,r)\geq v(g_{b})r^{n}$ for all radii $r>0$ and
points $x\in N$ so that well-known Gaussian estimates established by Li-Yau
[LY86] can be used.

Reasoning as in (LABEL:dealing-with-non-lin-Ric-wei) would have led us to
estimate a priori a term such as

$\int_{0}^{t}\|(g(s)-g_{b})\ast\nabla^{g_{b},2}\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}\,ds.$

In particular, we are left with estimating

$\sup_{s\in[0,T)}\|(g(s)-g_{b})\|_{C^{0}_{2}}\int_{0}^{t}\|\nabla^{g_{b},2}\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau}}\,ds$

from above and uniformly in time. This is fine by Lemma 2.14 as long as
$\tau\geq 2$. Such a condition is always satisfied if $n\geq 6$ or if $n=5$ by
restricting $\tau$. However, we are stuck in dimension $4$ since
$\tau\in(1,2)$.

We end this section by establishing an a priori $C^{2,\alpha}_{\tau}$-estimate
on $g(t)-g_{b}$, $t\in[0,T)$ as long as $g(t)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$.

######  Lemma 2.17 (A priori weighted $C^{2,\alpha}_{\tau}$ estimate).

Let $(N^{n},g_{b})$, $n\geq 4$, be an ALE Ricci flat metric. Let
$\tau\in(\frac{n-2}{2},n-2)$, $\alpha\in(0,1)$ and let $(g(t))_{t\in[0,T)}$ be
a solution to the Ricci flow in a neighborhood
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ with uniformly bounded covariant
derivatives, i.e. $\sup_{t\in[0,T)}\|\nabla^{g(t)}\mathop{\rm
Rm}\nolimits(g(t))\|_{C^{0}}\leq C_{k}$, $k\geq 1$, and such that

$\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\leq\frac{1}{2n}.$ (2.52)

Then for $\eta\in(0,1)$ and
$\alpha\in\left(0,\min\left\\{1,\tau-1,n-2-\tau\right\\}\right)$:

$\begin{split}\|g(t)-g(s)\|_{C^{2,\alpha}_{\tau}}\leq
C_{\eta}\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0,\alpha}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{1-\eta}_{C^{0}}\,dt^{\prime}\right),\quad
0\leq s<t<T,\end{split}$ (2.53)

for some time-independent positive constant
$C_{\eta}=C(n,g_{b},\varepsilon,\eta)$.

###### Proof.

By elliptic regularity, it is sufficient to establish an a priori bound on the
$C^{0,\alpha}_{\tau+2}$-norm of $\Delta_{g_{b}}(g(t)-g_{b})$ or equivalently
$\Delta_{g(t)}(g(t)-g_{b})$ since $g(t)$ is $\varepsilon$-close to $g_{b}$ in
the $C^{2,\alpha}_{\tau}$-topology. Indeed, observe that if
$h(t):=g(t)-g_{b}$, then schematically,

$\Delta_{g(t)}h(t)-\Delta_{g_{b}}h(t)=\nabla^{g_{b},2}h(t)\ast
h(t)+\nabla^{g_{b}}h(t)\ast\nabla^{g_{b}}h(t),$

which implies

$\|\Delta_{g_{b}}h(t)\|_{C^{0,\alpha}_{\tau+2}}\leq\|\Delta_{g(t)}h(t)\|_{C^{0,\alpha}_{\tau+2}}+c(n)\|h(t)\|_{C^{2,\alpha}_{\tau}}\|h(t)\|_{C^{1}}.$

In particular, (A.21) from the proof of Lemma A.7 leads to the following
estimate on the $C^{0}_{\tau+2}$ estimate:

$\begin{split}\|\Delta_{g(t)}(g(t)-g(s))\|_{C^{0}_{\tau+2}}\leq\,&2\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau+2}}+2\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0}_{\tau+2}}\\\
&+\sup_{t^{\prime}\in[s,t]}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0}_{2}}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}\\\
&+\sum_{k=0}^{2}\int_{s}^{t}\|\nabla^{g(t^{\prime}),k}(g(t^{\prime})-g(s))\|_{C^{0}_{2}}\|\nabla^{g(t^{\prime}),2-k}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}.\end{split}$ (2.54)

Invoking Lemmata 2.14 and 2.15, the proof of (2.53) with $\alpha=0$ ends
provided $\tau\geq 2$. Indeed, in order to bound the norms
$\|\nabla^{g(t^{\prime}),k}(g(t^{\prime})-g(s))\|_{C^{0}_{2}}$, $k=0,1,2$,
uniformly in time, this requires either $n\geq 6$ or $n=5$ by restricting
$\tau$ accordingly: this phenomenon echoes Remark 2.16.

Arguing as in (2.54) leads to the desired expected estimate on
$\|\Delta_{g(t)}(g(t)-g_{b})\|_{C^{0,\alpha}_{\tau+2}}$ by using Lemma (A.7)
instead. Indeed, we arrive at the following estimate:

$\begin{split}\|\Delta_{g(t)}(g(t)-g(s))\|_{C^{0,\alpha}_{\tau+2}}\leq\,&C\left(\|\mathop{\rm
Ric}\nolimits(g(s)\|_{C^{0,\alpha}_{\tau+2}}+\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0,\alpha}_{\tau+2}}\right)\\\ &+C\sup_{s\leq
t^{\prime}\leq t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}_{\tau+2}}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}}\,dt^{\prime}\\\ &+C\sup_{s\leq
t^{\prime}\leq
t}\|\nabla^{g(t^{\prime}),2}(g(t^{\prime})-g(s))\|_{C^{0,\alpha}_{\tau+2}}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}}\,dt^{\prime}\\\ &+C\sup_{s\leq
t^{\prime}\leq
t}\|\nabla^{g(t^{\prime})}(g(t^{\prime})-g(s))\|_{C^{0,\alpha}_{\tau+1}}\int_{s}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C_{1}^{0,\alpha}}\,dt^{\prime}\\\
\leq\,&C_{\eta}\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{2,\alpha}_{\tau+2}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}^{1-\eta}\,dt^{\prime}\right)\\\
&+C\int_{s}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C_{1}^{0,\alpha}}\,dt^{\prime},\end{split}$
(2.55)

if $\alpha\in(0,\min\\{1,n-2-\tau\\})$ and $\eta\in(0,1)$. Here we have
invoked Lemma 2.15 in the second inequality which explains the restriction on
$\alpha$. Now, if $\alpha\in(0,\min\\{1,\tau-1\\})$, a similar reasoning that
led to (2.50) in the proof of Lemma 2.15 gives an estimate of the last
integral on the righthand side of (2.55) as follows if
$\gamma\in\left(0,1-\frac{1+\alpha}{\tau}\right)$ and $x\in N$:

$\begin{split}\rho_{g_{b}}(x)^{1+\alpha}\int_{s}^{t}&|[\nabla^{g(s)}\mathop{\rm
Ric}\nolimits(g(s))]_{C^{\alpha}}|(x)\,ds\leq\,C_{\alpha,\gamma}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}^{\frac{1+\alpha}{\tau}}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{1-\gamma-\frac{1+\alpha}{\tau}}_{C^{0}}\,dt^{\prime}\\\
\leq\,&C_{\alpha,\gamma}\left(\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau}}\,dt^{\prime}\right)^{\frac{1+\alpha}{\tau}}\cdot\left(\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{1-\gamma\frac{\tau}{\tau-1-\alpha}}_{C^{0}}\,dt^{\prime}\right)^{1-\frac{1+\alpha}{\tau}}.\end{split}$

The case $\alpha=0$ is handled similarly. Therefore, if $\eta\in(0,1)$, an
application of Lemma 2.14 leads us to,

$\begin{split}\int_{s}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C_{1}^{0,\alpha}}\,dt^{\prime}\leq&\,C_{\alpha,\eta}\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\right)\\\
&+C_{\alpha,\eta}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}^{1-\eta}\,dt^{\prime}\\\
\leq&\,C_{\alpha,\eta}\left(\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}_{\tau}}+\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{1-\eta}_{C^{0}}\,dt^{\prime}\right)\end{split}$

in case $\alpha\in(0,\min\\{1,n-2-\tau,\tau-1\\})$ and this ends the proof of
the desired estimate.

∎

##  3\. Stability of Ricci-flat ALE metrics

###  3.1. A stability result

Let us now use the Łojasiewicz inequality (0.2) to study the stability of
Ricci-flat ALE metrics. We will say that a Ricci-flat ALE metric
$(N^{n},g_{b})$ is stable if it is a local maximizer of
$\lambda_{\operatorname{ALE}}$, and unstable otherwise. From now on, we make
the following assumption.

In a $C^{2,\alpha}_{\tau}(g_{b})$-neighborhood
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$ of $g_{b}$, an
$L^{2}$-Łojasiewicz inequality is satisfied: for any metric $g$ in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$, we have

$|\lambda_{\operatorname{ALE}}(g)|^{2-\theta}\leq
C\|\nabla\lambda_{\operatorname{ALE}}(g)\|_{L^{2}(g_{b})}^{2},$ (3.1)

for some $\theta\in(0,1)$.

The purpose of this section is to give a proof of the following stability
result: in the stable case, just like in [Has12], [HM14] or [KP20], the Ricci
flow converges to a Ricci-flat metric at a particular rate.

######  Theorem 3.1.

Let $n\geq 4$ and $\tau\in(\frac{n-2}{2},n-2)$. Let
$\alpha\in\left(0,\min\left\\{1,\tau-1,n-2-\tau\right\\}\right)$. Let
$(N^{n},g_{b})$ be a stable Ricci-flat ALE metric such that Inequality 3.1
holds on a neighborhood $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$ with
exponent $\theta\in(0,1)$.

Then for every $\varepsilon\in(0,\varepsilon_{\L})$, there exists $\delta>0$
such that the Ricci flow starting at any metric in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\delta)$ stays in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ and converges to a Ricci-flat
metric $g_{\infty}$ in $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ in the
$C^{2,\alpha^{\prime}}_{\tau^{\prime}}$-topology for any
$\tau^{\prime}\in(\frac{n-2}{2},\tau)$ and $\alpha^{\prime}\in(0,\alpha)$.

Moreover there exists a positive constant $C=C(n,g_{b},\varepsilon,\theta)$
such that

$\|g(t)-g_{\infty}\|_{C^{0}}\leq Ct^{-\frac{\theta}{2(1-\theta)}},\quad t\geq
1,$ (3.2)

and

$\int_{0}^{+\infty}\|\mathop{\rm Ric}\nolimits(g(s))\|_{C^{0}}+\|\mathop{\rm
Ric}\nolimits(g(s))\|_{L^{2}}\,ds\leq
C(n,g_{b},\varepsilon,\theta)\left(\delta+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}}\right).\\\
$ (3.3)

######  Remark 3.2.

The convergence in which Theorem 3.1 takes place is reminiscent of the
convergence result obtained in [Li18, Theorem $5.1$]. However, unless the
background Ricci flat metric $(N^{n},g_{b})$ is flat, the solution
$(N^{n},g(t))_{t\geq 0}$ we provide is Type IIb, i.e.

$\limsup_{t\rightarrow+\infty}t|\mathop{\rm
Rm}\nolimits(g(t))|_{g(t)}=+\infty.$

Because of this fact, it is unclear if one can get a convergence for all
rescaled covariant derivatives of the metric as in [Li18, Theorem $5.1$].

To end this section, let us discuss the need for Gaussian bounds on the heat
kernel given by Theorem 2.6. In Proposition 3.5, we start by proving an a
priori $C^{0}$ estimate on the distance of a Ricci flow $(g(t))_{t\in[0,T)}$
to the origin $g_{b}$ lying in a small neighborhood
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$.

At this stage, one is tempted to use appropriate interpolation inequalities
together with the a priori $L^{2}$ bound on $\mathop{\rm Ric}\nolimits(g(t))$,
$t\in[0,T)$, established in Lemma 3.6 to get such a time-independent a priori
$C^{0}_{\tau}$ bound on $g(t)-g_{b}$, $t\in[0,T)$. Let us apply the Gagliardo-
Nirenberg interpolation inequalities [Aub82, Theorem $3.70$, Chapter $3$]
which can be adapted to $(N^{n},g_{b})$, to the tensor $\partial_{t}g$ with
$p=\infty$, $j=0$, $r=\infty$, $q=2$ and $m\geq 1$ to get:

$\begin{split}\rho_{g_{b}}(x)^{\tau}\|\partial_{t}g\|_{C^{0}(B_{g_{b}}(x,\rho_{g_{b}}(x)/4))}&\leq
C(n,g_{b})\|\partial_{t}g\|_{L^{2}}^{1-a}\cdot\|\partial_{t}g\|_{C^{m}_{\tau+2}}^{a}\cdot\rho_{g_{b}}(x)^{\tau-a(m+\tau+2)},\end{split}$

where $a\in(0,1)$ is such that $(2m+n)a=n$ and if $(g(t))_{t\in[0,T)}$ is
assumed to stay in a fixed neighborhood
$B_{C^{m+2,\alpha}_{\tau}}(g_{b},\varepsilon)$ of $g_{b}$. This estimate
already shows that it asks for too much regularity of the solution. Moreover,
one needs to ensure the exponent $\tau-a(m+\tau+2)$ to be non-positive which
constrains $\tau$ to lie below $\frac{n}{2}\cdot\frac{m+2}{m}$. Worse, the
application of Lemma 3.6 requires the exponent $\eta$ (which equals $a$ here)
to be strictly less than $\frac{\theta}{2-\theta}$, this in turn restricts the
range of the exponent $\theta$ in the Łojasiewicz inequality from Theorem 1.12
which is uncheckable in general.

For all these reasons, we somewhat proceed more directly by using the heat
kernel estimates from Section 2.1 via Duhamel’s formula.

###  3.2. Proof of Theorem 3.1

We first show that given $\varepsilon$ as in the statement of Theorem 3.1,
there exists $\delta>0$ such that the Ricci flow starting at any metric $g(0)$
in $B_{C^{2,\alpha}_{\tau}}(g_{b},\delta)$ stays in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ and exists for all time.

Let $g(0)\in B_{C^{2,\alpha}_{\tau}}(g_{b},\delta)$ with $\delta<\varepsilon$
to be constrained later and let
$(N^{n},g(t))_{t\in[0,T_{\operatorname{Shi}}]}$ be Shi’s solution [Shi89a]
with

$T_{\operatorname{Shi}}:=\frac{\varepsilon(n)}{\sup_{N}|\mathop{\rm
Rm}\nolimits(g(0))|_{g(0)}}\geq\frac{\varepsilon(n)}{C(n,g_{b},\varepsilon)}=:T(n,g_{b})>0,$
(3.4)

which exists since $g(0)$ is $\varepsilon$-close to $g_{b}$ in the
$C^{2,\alpha}_{\tau}$-topology.

We define the maximal time of existence of this solution to the Ricci flow
with respect to the $C^{2,\alpha}_{\tau}$-topology as follows:

$T_{\operatorname{max}}:=\sup\left\\{T>0\,|\,\quad g(t)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon),\quad\forall t\in[0,T)\right\\}.$

We start with the following proposition:

######  Proposition 3.3.

There exists $\delta(n,g_{b},\varepsilon)>0$ such that for
$0<\delta\leq\delta(n,g_{b},\varepsilon)$, one has $T_{\operatorname{max}}\geq
T(n,g_{b})>0$ and the set

$\left\\{T>0\,|\,\quad\forall t\in[0,T),\quad g(t)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)\right\\},$

is open.

######  Remark 3.4.

The proof of Proposition 3.3 does not use the stability of $(N^{n},g_{b})$.

###### Proof of Proposition 3.3.

We only prove the fact that $T_{\operatorname{max}}\geq T(n,g_{b})$, since the
proof of the openness of the set of solutions is very similar and will
therefore be omitted.

According to Propositions A.5 and A.6,

$\begin{split}&\|\mathop{\rm Rm}\nolimits(g(t))\|_{C^{0}}\leq 2\|\mathop{\rm
Rm}\nolimits(g(0))\|_{C^{0}}\leq C(n,g_{b}),\\\ &\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0}}\leq e^{C(n,g_{b})\cdot t}\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0}}\leq C(n,g_{b})\delta,\end{split}$ (3.5)

for $t\in[0,T_{\operatorname{Shi}}]$. In particular, if $t\in[0,T(n,g_{b})]$,

$\|g(t)-g_{b}\|_{C^{0}}\leq 2\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}}\,ds+\|g(0)-g_{b}\|_{C^{0}}\leq
C(n,g_{b})\delta<\varepsilon,$

if $\delta$ is chosen small enough. Moreover, we choose $\delta$ small enough
so that $\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}}\,ds\leq\frac{1}{2n}$. Thanks to this choice,
Lemma 2.12 is applicable and gives us $\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0,\alpha}_{\tau+2}}\leq C(n,g_{b})\delta$ for
$t\in[0,T(n,g_{b})]$. By integrating in time this inequality and by reducing
$\delta$ once more if necessary, one gets for $t\in[0,T(n,g_{b})]$ that
$\|g(t)-g_{b}\|_{C^{0}_{\tau}}<\varepsilon$ (we actually get a stronger decay
in space but we do not use this fact as it will not be preserved for large
time).

Finally, we invoke Lemma A.7 to get an a priori bound on the full
$C^{2,\alpha}_{\tau}$-norm. Indeed, with $0=:s<t\leq T(n,g_{b})$, Lemma A.7
leads to:

$\begin{split}\|\Delta_{g(t)}(g(t)-g(0))\|_{C^{0,\alpha}_{\tau+2}}\leq\,&\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0,\alpha}_{\tau+2}}+\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0,\alpha}_{\tau+2}}\\\ &+\int_{0}^{t}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}_{\tau+2}}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}}\,dt^{\prime}\\\
&+\int_{0}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}_{1}}\|g(t^{\prime})-g(0)\|_{C^{2,\alpha}_{\tau}}\,dt^{\prime}\\\
&+\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}}\|g(t^{\prime})-g(0)\|_{C^{2,\alpha}_{\tau}}\,dt^{\prime}\\\
\leq\,&C(n,g_{b})\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0,\alpha}_{\tau+2}}+C(n,g_{b})t\sup_{t^{\prime}\in[0,t]}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}}\\\
&+\int_{0}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}_{1}}\|g(t^{\prime})-g(0)\|_{C^{2,\alpha}_{\tau}}\,dt^{\prime}\\\
\leq\,&C(n,g_{b})\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0,\alpha}_{\tau+2}}\\\
&+\sup_{t^{\prime}\in[0,t]}\|g(t^{\prime})-g(0)\|_{C^{2,\alpha}_{\tau}}\int_{0}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0,\alpha}_{1}}\,dt^{\prime}.\end{split}$

Here we have used Lemma 2.12 in the second inequality. Observe that thanks to
[(A.15), Proposition A.5],

$\begin{split}\int_{0}^{t}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{1}}\,dt^{\prime}\leq\,&C\int_{0}^{t}\frac{\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0}_{1}}}{\sqrt{t^{\prime}}}\,dt^{\prime}\\\
\leq\,&C\|\mathop{\rm Ric}\nolimits(g(0))\|_{C^{0}_{\tau+2}}\\\
\leq\,&C\|g(0)-g_{b}\|_{C^{2,\alpha}_{\tau}},\end{split}$ (3.6)

for some positive constant $C=C(n,g_{b},\varepsilon)$ that may vary from line
to line.

Similarly,

$\begin{split}\int_{0}^{t}\rho_{g_{b}}^{1+\alpha}&[\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))]_{C^{\alpha}}\,dt^{\prime}\\\
\leq\,&C\int_{0}^{t}\rho_{g_{b}}^{1+\alpha-(\tau+2)}\|\nabla^{g(t^{\prime})}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau+2}}^{1-\alpha}\|\nabla^{g(t^{\prime}),2}\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}_{\tau+2}}^{\alpha}\,dt^{\prime}\\\
\leq\,&C\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0}_{\tau+2}}\int_{0}^{t}\frac{1}{t^{\prime\frac{1-\alpha}{2}+\alpha}}\,dt^{\prime}\\\
\leq\,&C\|\mathop{\rm Ric}\nolimits(g(0))\|_{C^{0}_{\tau+2}}\\\
\leq\,&C\|g(0)-g_{b}\|_{C^{2,\alpha}_{\tau}},\end{split}$ (3.7)

for some positive constant $C=C(n,g_{b},\varepsilon)$ that may vary from line
to line. Here we have used that $1+\alpha-(\tau+2)\leq 0$ since $0<\alpha<1$
and $\tau$ is positive.

Therefore, as an intermediate conclusion, we obtain:

$\begin{split}\|\Delta_{g(t)}(g(t)-g(0))\|_{C^{0,\alpha}_{\tau+2}}\leq\,&C(n,g_{b},\varepsilon)\delta\left(1+\sup_{t^{\prime}\in[0,t]}\|g(t^{\prime})-g(0)\|_{C^{2,\alpha}_{\tau}}\right).\end{split}$
(3.8)

Now, observe that if $h(t):=g(t)-g(0)$, then schematically,

$\Delta_{g(t)}h(t)-\Delta_{g(0)}h(t)=\nabla^{g(0),2}h(t)\ast
h(t)+\nabla^{g(0)}h(t)\ast\nabla^{g(0)}h(t),$

which implies

$\|\Delta_{g(0)}h(t)\|_{C^{0,\alpha}_{\tau+2}}\leq\|\Delta_{g(t)}h(t)\|_{C^{0,\alpha}_{\tau+2}}+C(n,g_{b})\|h(t)\|_{C^{2,\alpha}_{\tau}}\|h(t)\|_{C^{1}}.$
(3.9)

Plugging (3.8) in the previous estimate (3.9) leads to:

$\|\Delta_{g(0)}h(t)\|_{C^{0,\alpha}_{\tau+2}}\leq
C(n,g_{b},\varepsilon)\delta\left(1+\sup_{t^{\prime}\in[0,t]}\|h(t^{\prime})\|_{C^{2,\alpha}_{\tau}}\right)+C(n,g_{b})\|h(t)\|_{C^{2,\alpha}_{\tau}}\|h(t)\|_{C^{1}}.$

According to elliptic Schauder estimate applied to the background initial
metric $g(0)$, one gets $\|h(t)\|_{C^{2,\alpha}_{\tau}}\leq
C(n,g_{b},\varepsilon)\left(\|\Delta_{g(0)}h(t)\|_{C^{0,\alpha}_{\tau+2}}+\|h(t)\|_{C^{0}_{\tau}}\right)$
which leads to:

$\|h(t)\|_{C^{2,\alpha}_{\tau}}\leq
C(n,g_{b},\varepsilon)\delta\left(1+\sup_{t^{\prime}\in[0,t]}\|h(t^{\prime})\|_{C^{2,\alpha}_{\tau}}\right)+C(n,g_{b})\|h(t)\|_{C^{2,\alpha}_{\tau}}\|h(t)\|_{C^{1}}.$
(3.10)

Notice that (3.6) and (3.7) imply a $C^{1,\alpha}$-estimate on $g(t)-g(0)$ by
integrating over $[0,T(n,g_{b})]$:
$\sup_{t^{\prime}\in[0,T(n,g_{b})]}\|h(t^{\prime})\|_{C^{1,\alpha}}\leq
C(n,g_{b},\varepsilon)\delta.$

We are then in a position to conclude since this previous fact combined with
(3.10) imply if $\delta\leq\delta(n,g_{b},\varepsilon)$:

$\|h(t)\|_{C^{2,\alpha}_{\tau}}\leq
C(n,g_{b},\varepsilon)\delta\left(1+\sup_{t^{\prime}\in[0,t]}\|h(t^{\prime})\|_{C^{2,\alpha}_{\tau}}\right),\quad
t\in[0,T(n,g_{b})].$ (3.11)

Considering the function in time
$\sup_{t^{\prime}\in[0,t]}\|h(t^{\prime})\|_{C^{2,\alpha}_{\tau}}$, one gets,
by choosing $\delta\leq\delta(n,g_{b},\varepsilon)$ sufficiently small that
$\|h(t)\|_{C^{2,\alpha}_{\tau}}\leq C(n,g_{b},\varepsilon)\delta$ for all
$t\in[0,T(n,g_{b})]$ which by the triangular inequality leads to the expected
result, i.e. $g(t)-g_{b}\in B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ for
all $t\in[0,T(n,g_{b})]$. ∎

The next result consists in showing a priori estimates on the
$C^{2,\alpha}_{\tau}$-norm of our solution which are time-independent. This is
where the Łojasiewciz inequality comes into play.

######  Proposition 3.5.

The set

$\left\\{T>0\,|\,\quad\forall t\in[0,T),\quad g(t)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)\right\\},$

is closed. More precisely, there exist time-independent positive constants
$C=C(n,g_{b},\varepsilon,\theta)$ and $C=C(n,g_{b},\varepsilon,\theta,\eta)$,
$\eta\in\left(0,\frac{\theta}{2-\theta}\right)$, such that for $t\in[0,T)$:

$\displaystyle\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}$ $\displaystyle\leq$
$\displaystyle
C(n,g_{b},\varepsilon,\theta)\left(\delta+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}}\right),$
(3.12) $\displaystyle\|g(t)-g(0)\|_{C^{0}_{\tau}}$ $\displaystyle\leq$
$\displaystyle
C(n,g_{b},\varepsilon,\theta)\left(\delta+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}}\right),$
(3.13) $\displaystyle\|g(t)-g(0)\|_{C^{2,\alpha}_{\tau}}$ $\displaystyle\leq$
$\displaystyle
C(n,g_{b},\varepsilon,\theta,\eta)\left(\delta^{1-\eta}+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}(1+\eta)-\eta}\right).$
(3.14)

Before we prove Proposition 3.5, we establish one more crucial lemma which
gives an a priori $L^{2}$ control on the distance of a Ricci flow from the
origin given by a stable Ricci flat ALE metric:

######  Lemma 3.6 (A priori $L^{2}$ estimate for the Ricci flow).

Let $(N^{n},g_{b})$ be a stable ALE Ricci flat metric such that Inequality 3.1
holds on a neighborhood $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$ with
exponent $\theta\in(0,1)$. Let $(g(t))_{t\in[0,T)}$ be a solution to the Ricci
flow in $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$,
$\varepsilon<\varepsilon_{\L}$. Then, one has the following decay in time:

$\begin{split}C^{-1}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))+&\nabla^{g(t^{\prime}),2}f_{g(t^{\prime})}\|_{L^{2}}^{2}\,dt^{\prime}\leq|\lambda_{\operatorname{ALE}}(g(s))|\leq\frac{|\lambda_{\operatorname{ALE}}(g(0))|}{\left(1+C|\lambda_{\operatorname{ALE}}(g(0))|^{1-\theta}\cdot
s\right)^{1+\beta_{\theta}}},\\\ &\quad 0\leq
s<t<T,\quad\beta_{\theta}:=\frac{\theta}{1-\theta},\end{split}$ (3.15)

for some positive constant $C=C(n,g_{b},\varepsilon,\theta)$ independent of
time and where the $L^{2}$ norm is understood with respect to the weighted
measure $e^{-f_{g(t^{\prime})}}d\mu_{g(t^{\prime})}$.

In particular, one has the following uniform energy bound if $0\leq s<t<T$:

$\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{L^{2}}^{2}+\|\nabla^{g(t^{\prime}),2}f_{g(t^{\prime})}\|^{2}_{L^{2}}\,dt^{\prime}\leq\frac{C|\lambda_{\operatorname{ALE}}(g(0))|}{\left(1+C|\lambda_{\operatorname{ALE}}(g(0))|^{1-\theta}\cdot
s\right)^{1+\beta_{\theta}}}.$ (3.16)

Finally, if $\eta\in\left[0,\frac{\theta}{2-\theta}\right)$, one has

$\begin{split}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{L^{2}}^{1-\eta}+\|\nabla^{g(t^{\prime}),2}f_{g(t^{\prime})}\|_{L^{2}}^{1-\eta}\,dt^{\prime}\leq
C|\lambda_{\operatorname{ALE}}(g(s))|^{\frac{\theta}{2}(1+\eta)-\eta},\quad
0\leq s<t<T,\end{split}$ (3.17)

for some positive constant $C=C(n,g_{b},\varepsilon,\theta,\eta)$.

######  Remark 3.7.

The right-hand side of (3.16) also tends to $0$ as $s\to\infty$ for some
negative values of $\theta$. Recall that up to a second order error, we have
$\lambda_{\operatorname{ALE}}(g)\sim\|g-g_{b}\|^{2}_{H^{1}_{n/2-1}}$ if
$g-g_{b}\perp\ker_{L^{2}}L_{g_{b}}$. A natural question would be: can the
Ricci flows considered in the proof of Theorem 3.1 still converge in
$H^{1}_{n/2-1}$ if a sufficiently good $L^{2}$-Łojasiewicz inequality does not
hold?

###### Proof.

Observe first that by definition of the Ricci flow together with the first
variation of $\lambda_{\operatorname{ALE}}$ computed in [(1.9), Proposition
1.9], one has for $t>s\geq 0$,

$\begin{split}2\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))+\nabla^{g(t^{\prime}),2}f_{g(t^{\prime})}\|_{L^{2}(e^{-f_{g(t)}}d\mu_{g(t)})}^{2}\,dt^{\prime}&=\lambda_{\operatorname{ALE}}(g(t))-\lambda_{\operatorname{ALE}}(g(s))\\\
&\leq-\lambda_{\operatorname{ALE}}(g(s))\leq|\lambda_{\operatorname{ALE}}(g(0))|.\end{split}$
(3.18)

Here, we have used the fact that
$0=\lambda_{\operatorname{ALE}}(g_{b})\geq\lambda_{\operatorname{ALE}}(g)$ for
all $g\in B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ since $g_{b}$ is a local
maximizer of $\lambda_{\operatorname{ALE}}$ by assumption. This proves the
first part of (3.15) since the measures $e^{-f_{g(t)}}d\mu_{g(t)}$ and
$d\mu_{g(t)}$ are uniformly equivalent in time and space.

By using the Łojasiewicz inequality for ALE metrics given by Inequality 3.1
with exponent $\theta$, one gets:

$\begin{split}\frac{d}{dt}(-\lambda_{\operatorname{ALE}}(g(t)))\leq-C(-\lambda_{\operatorname{ALE}}(g(t)))^{2-\theta},\quad
t\in(0,T),\end{split}$ (3.19)

for some positive constant $C$ independent of time.

Integrating this differential inequality leads to:

$(-\lambda_{\operatorname{ALE}}(g(t)))\leq\frac{|\lambda_{\operatorname{ALE}}(g(0))|}{\left(1+C|\lambda_{\operatorname{ALE}}(g(0))|^{1-\theta}\cdot
t\right)^{\frac{1}{1-\theta}}},\quad 0\leq t<T,$ (3.20)

for some positive constant $C$ independent of time. This gives us the full
estimate (3.15). The a priori $L^{2}$-estimate (3.16) follows by invoking the
previous bound (3.15) together with Proposition 2.8.

Now, on the one hand, consider the function
$t\rightarrow(-\lambda_{\operatorname{ALE}}(g(t)))^{\gamma}$ for some positive
constant $\gamma$ to be constrained later. Observe as in [HM14] that if
$\eta\in[0,\theta/(2-\theta))$ for some $\theta\in[0,1)$:

$\begin{split}\frac{d}{dt}(-\lambda_{\operatorname{ALE}}(g(t)))^{\gamma}&=-\gamma(-\lambda_{\operatorname{ALE}}(g(t)))^{\gamma-1}\frac{d}{dt}\lambda_{\operatorname{ALE}}(g(t))\\\
&=-2\gamma(-\lambda_{\operatorname{ALE}}(g(t)))^{\gamma-1}\|\mathop{\rm
Ric}\nolimits(g(t))+\nabla^{g(t),2}f_{g(t)}\|^{2}_{L^{2}}\\\
&=-2\gamma(-\lambda_{\operatorname{ALE}}(g(t)))^{\gamma-1}\|\mathop{\rm
Ric}\nolimits(g(t))+\nabla^{g(t),2}f_{g(t)}\|_{L^{2}}^{(1+\eta)+(1-\eta)}\\\
&\leq-C\gamma(-\lambda_{\operatorname{ALE}}(g(t)))^{\gamma-1+(1+\eta)\left(1-\frac{\theta}{2}\right)}\|\mathop{\rm
Ric}\nolimits(g(t))+\nabla^{g(t),2}f_{g(t)}\|_{L^{2}}^{1-\eta}\\\
&=-C\gamma\|\mathop{\rm
Ric}\nolimits(g(t))+\nabla^{g(t),2}f_{g(t)}\|_{L^{2}}^{1-\eta},\end{split}$
(3.21)

if

$\gamma:=\frac{\theta}{2}(1+\eta)-\eta>0.$

Here we have used the Łojasiewicz inequality for ALE metrics given by
Inequality 3.1 with exponent $\theta$ in the fourth line.

Integrating (3.21) in time, one gets the expected result, i.e.

$\begin{split}\int_{s}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))+\nabla^{g(t^{\prime}),2}f_{g(t^{\prime})}\|_{L^{2}}^{1-\eta}\,dt^{\prime}&\leq\frac{1}{C\gamma}|\lambda_{\operatorname{ALE}}(g(s))|^{\gamma},\quad
t>s\geq 0.\end{split}$ (3.22)

This holds true whenever this function is differentiable, i.e. as long as
$\lambda_{\operatorname{ALE}}(g(t))$ is not zero. Otherwise, one can adapt
this reasoning by considering
$t\rightarrow(-\lambda_{\operatorname{ALE}}(g(t))+\tilde{\varepsilon})^{\gamma}$
with $\tilde{\varepsilon}$ small and then we let $\tilde{\varepsilon}$ go to
$0$. This concludes the proof of (3.17) by invoking Proposition 2.8 once more.

∎

We are now in a good position to prove Proposition 3.5.

###### Proof of Proposition 3.5.

We proceed by establishing successive claims which lead to the expected
result. It is sufficient to prove the corresponding estimates for large time,
i.e. if $t\geq\frac{T(n,g_{b})}{2}$.

######  Claim 2.

For $t\in[0,T)$, (3.12) and the a priori $C^{0}_{\tau}$ estimate (3.13) hold
true.

###### Proof of Claim 2.

Let us check that the assumptions of Lemma 2.14 are satisfied. It is
sufficient to check condition (2.33) since we assume $g(t)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$. Since $g(t)\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ for all $t\in[0,T)$ then if
$r^{2}:=\frac{T(n,g_{b})}{2}$, it is straightforward to check that
$\int_{t-r^{2}}^{t}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds\leq\frac{1}{2}$ by considering $\varepsilon$
small enough. This fact lets us to use Proposition 2.9 to get for
$0<r^{2}:=\frac{T(n,g_{b})}{2}<t$:

$\begin{split}\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0}}\leq\,&C(n,g_{b},\varepsilon)\exp\left(c(n)\int_{t-r^{2}}^{t}\|\mathop{\rm
Rm}\nolimits(g(s))\|_{C^{0}}\,ds\right)\|\mathop{\rm
Ric}\nolimits(g(t-r^{2})\|_{L^{2}}\\\
\leq\,&C(n,g_{b},\varepsilon)\exp\left(C(n,g_{b},\varepsilon)r^{2}\right)\|\mathop{\rm
Ric}\nolimits(g(t-r^{2})\|_{L^{2}}\\\
\leq\,&C(n,g_{b},\varepsilon)\|\mathop{\rm
Ric}\nolimits(g(t-r^{2})\|_{L^{2}}.\end{split}$ (3.23)

Now, (3.23) and [(3.17), Lemma 3.6] with $s:=0<\frac{T(n,g_{b})}{2}\leq t<T$
and $\eta=0$ imply:

$\begin{split}\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\leq\,&\int_{0}^{\frac{T(n,g_{b})}{2}}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}+C(n,g_{b},\varepsilon,\theta)|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}}\\\
\leq\,&C(n,g_{b})\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0}}+C(n,g_{b},\varepsilon,\theta)|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}}\\\
\leq\,&C(n,g_{b},\varepsilon,\theta)\left(\delta+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}}\right),\end{split}$
(3.24)

where we have used Proposition A.6 in the second inequality. In particular, if
$\delta$ (and therefore $|\lambda_{\operatorname{ALE}}(g(0))|$) is chosen
small enough, then (2.33) is satisfied.

Applying [(2.35), Lemma 2.14] to $s:=0<t<T$ gives:

$\begin{split}\|g(t)-g(0)\|_{C^{0}_{\tau}}\leq\,&C(n,g_{b})\left(\|\mathop{\rm
Ric}\nolimits(g(0))\|_{C^{0}_{\tau+2}}+\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))\|_{C^{0}}\,ds^{\prime}\right)\\\
\leq\,&C(n,g_{b},\varepsilon,\theta)\left(\delta+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}}\right),\end{split}$

where we have used (3.24) in the second line. This concludes the proof of
Claim 2.

∎

######  Claim 3.

For $t\in[0,T)$ and $\eta\in\left(0,\frac{\theta}{2-\theta}\right)$,

$\|g(t)-g(0)\|_{C^{2,\alpha}_{\tau}}\leq
C(n,g_{b},\varepsilon,\theta,\eta)\left(\delta^{1-\eta}+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}(1+\eta)-\eta}\right).$
(3.25)

###### Proof of Claim 3.

It is sufficient to prove this claim if $t\geq\frac{T(n,g_{b})}{2}$.

In order to use Lemma 2.17, we first check that for each $k\geq 1$, there
exists some time-independent positive constant
$C_{k}=C(n,k,g_{b},\varepsilon)$ such that $\|\nabla^{g(t),k}\mathop{\rm
Rm}\nolimits(g(t))\|_{C^{0}}\leq C_{k}$ for $t\geq\frac{T(n,g_{b})}{2}$. Since
$g(t)\in B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ for all $t\in[0,T)$ by
assumption, $\sup_{t\in[0,T)}\|\mathop{\rm Rm}\nolimits(g(t))\|_{C^{0}}\leq
C(n,g_{b},\varepsilon)$. In particular, Proposition A.5 implies that
$\|\nabla^{g(t),k}\mathop{\rm Rm}\nolimits(g(t))\|_{C^{0}}\leq C_{k}$ for
$t=\frac{T(n,g_{b})}{2}$. Therefore, Lemma A.4 applies and guarantees that
such uniform-in-time bounds on the covariant derivatives of the curvature
tensor hold true for $t\geq\frac{T(n,g_{b})}{2}$.

Lemma 2.17 is therefore applicable: if $\eta\in(0,1)$,

$\begin{split}\|g(t)-g(0)\|_{C^{2,\alpha}_{\tau}}\leq\,&\left\|g(t)-g\left(\frac{T(n,g_{b})}{2}\right)\right\|_{C^{2,\alpha}_{\tau}}+\left\|g\left(\frac{T(n,g_{b})}{2}\right)-g(0)\right\|_{C^{2,\alpha}_{\tau}}\\\
\leq\,&C\|g(0)-g_{b}\|_{C^{2,\alpha}_{\tau}}\\\
&+C_{\eta}\left(\left\|\mathop{\rm
Ric}\nolimits\left(g\left(\frac{T(n,g_{b})}{2}\right)\right)\right\|_{C^{0,\alpha}_{\tau+2}}+\int_{\frac{T(n,g_{b})}{2}}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{1-\eta}_{C^{0}}\,dt^{\prime}\right).\end{split}$
(3.26)

Now, a similar argument based on [(3.17), Lemma 3.6] as in the proof of Claim
2 leads to:

$\|g(t)-g(0)\|_{C^{2,\alpha}_{\tau}}\leq
C\delta+C_{\eta}\left(\delta^{1-\eta}+|\lambda_{\operatorname{ALE}}(g(0))|^{\frac{\theta}{2}(1+\eta)-\eta}\right),$

under the restriction that $\eta\in\left(0,\frac{\theta}{2-\theta}\right)$.
This ends the proof of Claim 3. ∎

Claim 3 ends the proof of this proposition by choosing $\delta$ (and therefore
$\lambda_{\operatorname{ALE}}(g(0))$) so small so that the righthand side of
(3.25) is less than $\frac{\varepsilon}{2}$, say. ∎

Notice that Propositions 3.3 and 3.5 prove that
$T_{\operatorname{max}}=+\infty$, i.e. the solution to the Ricci flow
considered at the beginning of this Section 3.2 is immortal.

The last step to prove Theorem 3.1 is summarized in the following proposition.

######  Proposition 3.8.

Let $(N^{n},g_{b})$ be a stable Ricci-flat ALE metric such that Inequality 3.1
holds on a neighborhood $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$ with
exponent $\theta\in(0,1)$. Let $(g(t))_{t\in[0,+\infty)}$ be a solution to the
Ricci flow in $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$,
$\varepsilon<\varepsilon_{\L}$. Then there exists an ALE Ricci flat metric
$g_{\infty}\in B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ such that:

$\|g(t)-g_{\infty}\|_{C^{0}}\leq Ct^{-\frac{\theta}{2(1-\theta)}},\quad t\geq
1,$ (3.27)

for some positive constant $C=C(n,g_{b},\varepsilon,\theta)$.

######  Remark 3.9.

The decay in time for the $C^{2,\alpha^{\prime}}_{\tau^{\prime}}$-norms,
$\tau^{\prime}\in[0,\tau)$, $\alpha^{\prime}\in[0,\alpha)$, can be obtained by
interpolation between the $C^{0}$ and $C^{2,\alpha}_{\tau}$ norms together
with (3.27). The proof of such interpolation inequalities for Hölder norms can
be found for instance in [Kry96, Chapter $3$, Theorem $3.2.1$].

###### Proof of Proposition 3.8.

Let us show first that there exists a unique limit metric $g_{\infty}\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ in the
$C^{2,\alpha^{\prime}}_{\tau^{\prime}}$-topology for any
$\tau^{\prime}\in[0,\tau)$ and $\alpha^{\prime}\in[0,\alpha)$.

First of all, since we assume $(g(t))_{t\geq 0}$ to lie in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$, compactness of the embedding
$C^{2,\alpha}_{\tau}\hookrightarrow C^{2,\alpha^{\prime}}_{\tau^{\prime}}$,
for $\alpha^{\prime}\in[0,\alpha)$, $\tau^{\prime}\in[0,\tau)$ ensured by
Lemma 1.3, there is a sequence $(g(t_{i}))_{i}$ converging to a $2$-tensor
$g_{\infty}$ in $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ in the
$C^{2,\alpha^{\prime}}_{\tau^{\prime}}$-topology for any
$\tau^{\prime}\in[0,\tau)$ and $\alpha^{\prime}\in[0,\alpha)$. The limit
$2$-tensor is a Riemannian metric since it is $\varepsilon$-close to $g_{b}$.

Now, if $t\geq t_{i}\geq 1$, (3.23) in the proof of Claim 2 together with
[(3.17), Lemma 3.6] applied to $\eta=0$ gives the expected decay in time for
the $C^{0}$ norm:

$\|g(t)-g(t_{i})\|_{C^{0}}\leq 2\int_{t_{i}}^{t}\|\mathop{\rm
Ric}\nolimits(g(s))\|_{C^{0}}\,ds\leq Ct_{i}^{-\frac{\theta}{2(1-\theta)}}.$
(3.28)

Inequality (3.28) shows that the limit metric $g_{\infty}$ is unique, i.e. it
does not depend on the sequence $(t_{i})_{i}$. This also proves (3.27).

In order to estimate the decay in time for the $C^{0}$-norm of the Ricci
tensor, we make use of Proposition 2.10 with $s:=\frac{t}{2}$, $t:=t$ and
$p>\frac{n}{\tau+2}$ to get:

$\begin{split}\|\mathop{\rm
Ric}\nolimits(g(t))\|_{C^{0}}\leq\,&Ct^{-\frac{n}{2p}}\left\|\mathop{\rm
Ric}\nolimits\left(g\left(\frac{t}{2}\right)\right)\right\|_{L^{p}}+C\sup_{t^{\prime}\in[0,T)}\|\mathop{\rm
Rm}\nolimits(g(t^{\prime}))\|_{C^{0}}\int_{\frac{t}{2}}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\\\
\leq\,&Ct^{-\frac{n}{2p}}\left\|\mathop{\rm
Ric}\nolimits\left(g\left(\frac{t}{2}\right)\right)\right\|_{L^{p}}+C\int_{\frac{t}{2}}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\\\
\leq\,&Ct^{-\frac{n}{2p}}\left\|\mathop{\rm
Ric}\nolimits\left(g\left(\frac{t}{2}\right)\right)\right\|_{L^{p}}+Ct^{-\frac{\theta}{2(1-\theta)}},\end{split}$
(3.29)

where $C=C(n,g_{b},\varepsilon,\theta,p)$ is a time-independent positive
constant that may vary from line to line. Now, if $p:=2>\frac{n}{\tau+2}$ by
the choice of $\tau$ in the previous estimate, Proposition A.6 together with
[(3.17), Lemma 3.6] lead to:

$\begin{split}\left\|\mathop{\rm
Ric}\nolimits\left(g\left(\frac{t}{2}\right)\right)\right\|_{L^{2}}\leq\,&C\int_{\frac{t}{2}-1}^{\frac{t}{2}}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{L^{2}}\,dt^{\prime}\\\
\leq\,&Ct^{-\frac{\theta}{2(1-\theta)}},\end{split}$ (3.30)

if $t$ is large enough for some positive constant
$C=C(n,g_{b},\varepsilon,\theta)$.

Therefore, (3.29) and (3.30) imply the following decay for the $C^{0}$ norm of
the Ricci tensor for $t_{i}$ large enough:

$\|\mathop{\rm Ric}\nolimits(g(t))\|_{C^{0}}\leq
Ct^{-\frac{\theta}{2(1-\theta)}}.$ (3.31)

Moreover, (3.31) shows that $\mathop{\rm Ric}\nolimits(g_{\infty})=0$, i.e.
$g_{\infty}$ is a Ricci flat metric in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$. In particular, by [BKN89],
$g_{\infty}$ is an ALE Ricci flat metric.

Now, [(3.13), Proposition 3.5] ensures that Lemma 2.14 is applicable and for
$t\geq t_{i}\geq 1$,

$\begin{split}\|g(t)-g(t_{i})\|_{C^{0}_{\tau}}\leq\,&C(n,g_{b},\varepsilon)\left(\|\mathop{\rm
Ric}\nolimits(g(t_{i}))\|_{C^{0}_{\tau+2}}+\int_{t_{i}}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\right)\\\
\leq\,&C(n,g_{b},\varepsilon)\left(\|\mathop{\rm
Ric}\nolimits(g(t_{i}))\|_{C^{0}_{\tau+2}}+|\lambda_{\operatorname{ALE}}(g(t_{i}))|^{\frac{\theta}{2}}\right),\end{split}$
(3.32)

where $C=C(n,g_{b},\varepsilon,\theta)$ is a time-independent positive
constant. Here we have used [(3.15), Lemma 3.6] in the fourth line.

∎

###  3.3. Evolution of the Bianchi form, the scalar curvature and the mass

In this section, we investigate the evolution of the Bianchi one-form together
with that of the scalar curvature along a Ricci flow $(N^{n},g(t))_{t\geq 0}$
satisfying the assumptions of Theorem 3.1. Let us denote $g_{\infty}$ the
limit ALE Ricci-flat metric of the solution $g(t)$ as $t$ tends to $+\infty$.

Recall that the Bianchi one-form $B(t)$ of $g(t)$ and $g_{\infty}$ is defined
by

$B(t)_{i}:=B(g(t),\nabla^{g_{\infty}},g(t))_{i}=\frac{1}{2}g(t)^{kl}\left(\nabla^{g_{\infty}}_{k}g(t)_{il}+\nabla^{g_{\infty}}_{l}g(t)_{ik}-\nabla_{i}^{g_{\infty}}g(t)_{kl}\right),\quad
i=1,...,n.$ (3.33)

######  Remark 3.10.

Notice that the difference between the Bianchi one-form of $g(t)$ and
$g_{\infty}$ introduced by Kotschwar following Hamilton and deTurck, and the
Bianchi operator defined in (A.2) is neglectible in the sense that if
$h(t):=g(t)-g_{\infty}$,

$\begin{split}B(t)_{i}-B_{g(t)}(h(t))_{i}=\,&B(g(t),\nabla^{g_{\infty}},g(t))_{i}-B_{g(t)}(g(t)-g_{\infty})_{i}\\\
=\,&\frac{1}{2}g(t)^{kl}\left(\nabla^{g_{\infty}}_{k}g(t)_{il}+\nabla^{g_{\infty}}_{l}g(t)_{ik}-\nabla^{g_{\infty}}g(t)_{kl}\right)\\\
&-\mathop{\rm
div}\nolimits_{g_{\infty}}(h(t))_{i}+\frac{1}{2}\nabla^{g_{\infty}}\mathop{\rm
tr}\nolimits_{g_{\infty}}h(t)_{i}\\\
=\,&\frac{1}{2}g(t)^{kl}\left(\nabla^{g_{\infty}}_{k}h(t)_{il}+\nabla^{g_{\infty}}_{l}h(t)_{ik}-\nabla^{g_{\infty}}h(t)_{kl}\right)\\\
&-\mathop{\rm
div}\nolimits_{g_{\infty}}(h(t))_{i}+\frac{1}{2}\nabla^{g_{\infty}}\mathop{\rm
tr}\nolimits_{g_{\infty}}h(t)_{i}\\\
=\,&\frac{1}{2}\left(g(t)^{kl}-g_{\infty}^{kl}\right)\left(\nabla^{g_{\infty}}_{k}h(t)_{il}+\nabla^{g_{\infty}}_{l}h(t)_{ik}-\nabla^{g_{\infty}}h(t)_{kl}\right).\end{split}$

Therefore, $B(t)-B_{g(t)}(h(t))=g(t)^{-1}\ast h(t)\ast
g_{\infty}^{-1}\ast\nabla^{g_{\infty}}h(t)$ is quadratic in $h$ and its
derivatives.

Next, we recall the evolution equation satisfied by the Bianchi gauge along
the Ricci flow with a Ricci flat background metric $g_{\infty}$ given in
[Kot17]:

######  Lemma 3.11 (Kotschwar).

Let $(N^{n},g_{\infty})$ be a Ricci flat metric and let $(g(t))_{t\in[0,T)}$
be a Ricci flow on $N$. Then, the evolution equation satisfied by $B$ is
schematically:

$\partial_{t}B=g(t)^{-1}\ast B(t)\ast\mathop{\rm
Ric}\nolimits(g(t))+g(t)^{-1}\ast
g(t)^{-1}\ast\nabla^{g_{\infty}}(g(t)-g_{\infty})\ast\mathop{\rm
Ric}\nolimits(g(t)),$ (3.34)

where, if $S$ and $T$ are two tensors on $N$, $S\ast T$ means some weighted
sum of contractions of the tensor product of $S$ and $T$ with respect to the
background metric $g_{b}$ with coefficients bounded by universal constants.

Our main result is the following proposition on the decay of the Bianchi one-
form in space-time coordinates:

######  Proposition 3.12.

Let $n\geq 4$ and $\tau\in(\frac{n-2}{2},n-2)$. Let
$\alpha\in\left(0,\min\left\\{1,\tau-1,n-2-\tau\right\\}\right)$. Let
$(N^{n},g_{b})$ be a stable Ricci-flat ALE metric. Let
$(g(t))_{t\in[0,+\infty)}$ be a solution to the Ricci flow in
$B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$, for some $\varepsilon>0$ which
satisfies the conclusion of Theorem 3.1 and let $g_{\infty}\in
B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ be its limit Ricci-flat metric
$g_{\infty}$. Then for any positive $\eta$ small enough and any $k\geq 0$, we
have

$\|\nabla^{g_{\infty},k}B(t)\|_{C^{0}}\leq
C_{\eta,k}\,t^{-\frac{\theta}{1-\theta}+\eta},\quad t\geq 1,$ (3.35)

for some positive constant $C=C(n,g_{b},\varepsilon,\theta,\eta,k)$.

######  Remark 3.13.

In [Proposition 6.9, [KP20]], the decay of the Bianchi one-form obtained in
Proposition 3.12 is improved if the perturbation lies in $L^{p}\cap
L^{\infty}$ for $p<n$ small enough.

###### Proof.

From now on, the symbol $\lesssim$ (respectively $\gtrsim$) means " less than
or equal" (respectively " larger than or equal") up to a positive
multiplicative constant that depends on $n$, $g_{b}$, $k$, $\alpha$ and
$\varepsilon$ only. All the norms are understood with respect the metric
$g_{\infty}$. For the sake of clarity, we omit the dependence of the Levi-
Civita connection on $g_{\infty}$. By (3.34), the norm of the Bianchi gauge
$B(t)$ satisfies in the weak sense:

$\partial_{t}|B|\gtrsim-|\mathop{\rm Ric}\nolimits(g(t))||B(t)|-|\nabla
h(t)||\mathop{\rm Ric}\nolimits(g(t))|,\quad t>0.$

By Grönwall’s inequality, one gets pointwise in space:

$\begin{split}|B(t)|\gtrsim\,&|B(s)|\exp\left\\{-\int_{s}^{t}|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))|\,dt^{\prime}\right\\}\\\
&-\int_{s}^{t}\exp\left\\{-\int_{t^{\prime}}^{t}|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))|\,ds^{\prime}\right\\}|\nabla
h(t^{\prime})||\mathop{\rm
Ric}\nolimits(g(t^{\prime}))|\,dt^{\prime},\,\,t>s>0.\end{split}$

This implies, once we let $t$ tend to $+\infty$:

$\begin{split}|B(s)|\lesssim\,&\int_{s}^{+\infty}\exp\left\\{\int_{s}^{t^{\prime}}|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))|\,ds^{\prime}\right\\}|\nabla
h(t^{\prime})||\mathop{\rm Ric}\nolimits(g(t^{\prime}))|\,dt^{\prime}\\\
\lesssim\,&\exp\left\\{\int_{s}^{+\infty}|\mathop{\rm
Ric}\nolimits(g(s^{\prime}))|\,ds^{\prime}\right\\}\int_{s}^{+\infty}|\nabla
h(t^{\prime})||\mathop{\rm Ric}\nolimits(g(t^{\prime}))|\,dt^{\prime}\\\
\lesssim\,&\int_{s}^{+\infty}|\nabla h(t^{\prime})||\mathop{\rm
Ric}\nolimits(g(t^{\prime}))|\,dt^{\prime}.\end{split}$ (3.36)

where $C=C(n,g_{b},\varepsilon,\theta)$ is a time-independent positive
constant. Here, we have used Proposition 3.5 in the last inequality.

Now, according to the conclusion of Theorem 3.1,
$\|g(t)-g_{\infty}\|_{C^{0}}\leq Ct^{-\frac{\theta}{2(1-\theta)}}$ if $t\geq
1$. Since for any $k\geq 0$ and $t\geq 1$, $|\nabla^{g(t),k}\mathop{\rm
Rm}\nolimits(g(t))|_{g(t)}\leq C_{k}$, for some time-independent positive
constant $C_{k}$ by Lemma A.4, standard interpolation inequalities show that
the same decay in time holds true for higher covariant derivatives of
$g(t)-g_{\infty}$ at the expense of an arbitrary small error: for any positive
$\eta$ small enough and any $k\geq 0$, one gets
$\|\nabla^{k}(g(t)-g_{\infty})\|_{C^{0}}\leq
Ct^{-\frac{\theta}{2(1-\theta)}+\eta}$.

Therefore, (3.36) leads to the expected estimate (3.35) for $s\geq 2$:

$\begin{split}\|B(s)\|_{C^{0}}\lesssim\,&s^{-\frac{\theta}{2(1-\theta)}+\eta}\int_{s}^{+\infty}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{C^{0}}\,dt^{\prime}\\\
\lesssim\,&C_{\eta}s^{-\frac{\theta}{2(1-\theta)}+\eta}|\lambda_{\operatorname{ALE}}(g(s-1))|^{\frac{\theta}{2}}\\\
\lesssim\,&C_{\eta}s^{-\frac{\theta}{(1-\theta)}+\eta},\end{split}$

for any $\eta>0$ small enough. Here we have used Proposition 2.9 with
$0<r^{2}:=1<t$ together with [(3.17), Lemma 3.6]. The corresponding decay in
time on the covariant derivatives of $B(t)$ are obtained by interpolation.

∎

We continue by estimating the decay in time of the $C^{0}$-norm of the scalar
curvature. In case the scalar curvature at time $t=0$ is integrable, we also
show this property is preserved: this echoes the result of [DM07, Theorem $1$,
part B]. Our proof is based on heat kernel estimates established in Section
2.1.

######  Proposition 3.14.

Let $n\geq 4$ and $\tau\in(\frac{n-2}{2},n-2)$. Let
$\alpha\in\left(0,\min\left\\{1,\tau-1,n-2-\tau\right\\}\right)$. Let
$(N^{n},g_{b})$ be a stable Ricci-flat ALE metric such that Inequality 3.1
holds on a neighborhood $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon_{\L})$ with
exponent $\theta\in(0,1)$. Let $(g(t))_{t\in[0,+\infty)}$ be a solution to the
Ricci flow in $B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$,
$\varepsilon<\varepsilon_{\L}$, which satisfies the conclusion of Theorem 3.1
and let $g_{\infty}\in B_{C^{2,\alpha}_{\tau}}(g_{b},\varepsilon)$ be its
limit Ricci-flat metric.

Then,

$\|\mathop{\rm R}\nolimits_{g(t)}\|_{C^{0}}\leq
C\left(t^{-\frac{\theta}{2(1-\theta)}-\frac{n}{4}}+t^{-\frac{1}{1-\theta}}\right),\quad
t\geq 1,$ (3.37)

for some time-independent positive constant $C=C(n,g_{b},\varepsilon,\theta)$.
Moreover, if $\mathop{\rm R}\nolimits_{g(0)}\in L^{1}$ then $\mathop{\rm
R}\nolimits_{g(t)}\in L^{1}$ for every $t\geq 0$ and,

$\|\mathop{\rm R}\nolimits_{g(t)}\|_{L^{1}}\leq C\left(\|\mathop{\rm
R}\nolimits_{g(0)}\|_{L^{1}}+|\lambda_{\operatorname{ALE}}(g(0))|\right),\quad
t\geq 0,$ (3.38)

some time-independent positive constant $C=C(n,g_{b},\varepsilon,\theta)$.

###### Proof.

According to the evolution equation satisfied by the scalar curvature given in
[(A.7), Lemma A.3] together with Duhamel’s formula:

$\begin{split}\mathop{\rm R}\nolimits_{g(t)}(x)=\int_{N}&K(x,t,y,s)\mathop{\rm
R}\nolimits_{g(0)}(y)\,d\mu_{g(0)}(y)\\\
&+\int_{0}^{t}\int_{N}K(x,t,y,s)|\mathop{\rm
Ric}\nolimits(g(s))|^{2}_{g(s)}(y)\,d\mu_{g(s)}(y)ds,\quad t\geq 0,\quad x\in
N.\end{split}$ (3.39)

In particular, if $2\leq s:=\frac{t}{2}<t$, Theorem 2.6 ensures that,

$\begin{split}\|\mathop{\rm R}\nolimits_{g(t)}\|_{C^{0}}\leq\,&\sup_{x\in
N}\left\|K\left(x,t,\cdot,\frac{t}{2}\right)\right\|_{L^{2}}\|\mathop{\rm
R}\nolimits_{g\left(\frac{t}{2}\right)}\|_{L^{2}}+\int_{\frac{t}{2}}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{2}_{C^{0}}\,dt^{\prime}\\\
\leq\,&\frac{C}{t^{\frac{n}{4}}}\|\mathop{\rm
R}\nolimits_{g\left(\frac{t}{2}\right)}\|_{L^{2}}+\int_{\frac{t}{2}}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{2}_{C^{0}}\,dt^{\prime}\\\
\leq\,&\frac{C}{t^{\frac{n}{4}}}\|\mathop{\rm
R}\nolimits_{g\left(\frac{t}{2}\right)}\|_{L^{2}}+C\int_{\frac{t}{4}}^{t}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|^{2}_{L^{2}}\,dt^{\prime}\\\
\leq\,&\frac{C}{t^{\frac{n}{4}}}\|\mathop{\rm
R}\nolimits_{g\left(\frac{t}{2}\right)}\|_{L^{2}}+Ct^{-\frac{1}{1-\theta}}.\end{split}$
(3.40)

The second line is justified by Proposition 2.9 applied to $0<r^{2}:=1<t$
which implies that $\frac{t}{2}-1\geq\frac{t}{4}$ since $t\geq 4$. The last
line is obtained thanks to [(3.16), Lemma 3.6].

Now, Proposition A.6 together with [(3.17), Lemma 3.6] lead to:

$\begin{split}\|\mathop{\rm
R}\nolimits_{g\left(\frac{t}{2}\right)}\|_{L^{2}}\leq\,&n\left\|\mathop{\rm
Ric}\nolimits\left(g\left(\frac{t}{2}\right)\right)\right\|_{L^{2}}\\\
\leq\,&C\int_{\frac{t}{2}-1}^{\frac{t}{2}}\|\mathop{\rm
Ric}\nolimits(g(t^{\prime}))\|_{L^{2}}\,dt^{\prime}\\\
\leq\,&Ct^{-\frac{\theta}{2(1-\theta)}},\end{split}$ (3.41)

if $t\geq 2$ is large enough, for some time-independent positive constant
$C=C(n,g_{b},\varepsilon,\theta)$. Estimates (3.40) and (3.41) end the proof
of (3.37).

In order to prove (3.38), we observe that by considering (3.39), one obtains:

$\begin{split}\|\mathop{\rm
R}\nolimits_{g(t)}\|_{L^{1}}\leq\,&\int_{N}|\mathop{\rm
R}\nolimits_{g(t)}(x)|\,d\mu_{g(t)}(x)\\\ \leq\,&\sup_{y\in
N}\|K(\cdot,t,y,0)\|_{L^{1}}\|\mathop{\rm R}\nolimits_{g(0)}\|_{L^{1}}\\\
&+\int_{0}^{t}\sup_{y\in N}\|K(\cdot,t,y,s)\|_{L^{1}}\int_{N}|\mathop{\rm
Ric}\nolimits(g(s))|_{g(s)}^{2}(y)\,d\mu_{g(s)}(y)ds\\\
\leq\,&e^{\int_{0}^{t}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds}\left(\|\mathop{\rm
R}\nolimits_{g(0)}\|_{L^{1}}+\int_{0}^{t}\|\mathop{\rm
Ric}\nolimits(g(s))\|_{L^{2}}^{2}\,ds\right)\\\
\leq\,&Ce^{\int_{0}^{+\infty}\|\mathop{\rm
R}\nolimits_{g(s)}\|_{C^{0}}\,ds}\left(\|\mathop{\rm
R}\nolimits_{g(0)}\|_{L^{1}}+|\lambda_{\operatorname{ALE}}(g(0))|\,ds\right)\\\
\leq\,&C\left(\|\mathop{\rm
R}\nolimits_{g(0)}\|_{L^{1}}+|\lambda_{\operatorname{ALE}}(g(0))|\,ds\right),\end{split}$

for some time-dependent positive constant $C=C(n,g_{b},\varepsilon,\theta)$.

Here, we have used [(2.3), Proposition 2.1] in the third inequality together
with [(3.16), Lemma 3.6] in the penultimate line. Finally, [(3.3), Theorem