††X. D. was partially supported by the Simons Foundation. C.W. was partially
supported by the Fundamental Research Funds for the Central Universities.

# Perelman’s functionals on manifolds with non-isolated conical singularities

Xianzhe Dai  Department of Mathematics, University of California, Santa
Barbara CA93106, USA<EMAIL_ADDRESS>and  Changliang Wang  School of
Mathematical Sciences and Institute for Advanced Study, Tongji University,
Shanghai 200092, China<EMAIL_ADDRESS>
###### Abstract.

In this article, we define Perelman’s functionals on manifolds with non-
isolated conical singularities by starting from a spectral point of view for
the Perelman’s $\lambda$-functional. (Our definition of non-isolated conical
singularities includes isolated conical singularities.) We prove that the
spectrum of Schrödinger operator $-4\Delta+R$ on manifolds with non-isolated
conical singularities consists of discrete eigenvalues with finite
multiplicities, provided that scalar curvatures of cross sections of cones
have a certain lower bound. This enables us to define the $\lambda$-functional
on these singular manifolds, and further, to prove that the infimum of
$W$-functional is finite, with the help of some weighted Sobolev inequalities.
Furthermore, we obtain some asymptotic behavior of eigenfunctions and the
minimizer of the $W$-functional near the singularity, and a more refined
optimal partial asymptotic expansion for eigenfunctions near isolated conical
singularities. We also study the spectrum of $-4\Delta+R$ and Perelman’s
functionals on manifolds with more general singularities, i.e. the
$r^{\alpha}$-horn singularities which serve as prototypes of algebraic
singularities.

###### Key words and phrases:

Perelman functionals, non-isolated conical singularities, eigenvalues of
Schrödinger operator, singular potential, Sobolev inequality, weighted Sobolev
inequality.

##  1\. Introduction

Perelman’s functionals are some of the crucial ingredients introduced by
Perelman [Per02] in his famous work on Ricci flow. These functionals play
critical roles in his celebrated proof of Thurston’s geometrization conjecture
and Poincaré conjecture, as well as in later studies of the Ricci flow.

We briefly recall Perelman’s functionals on compact manifolds. Let $(M,g)$ be
a compact Riemannian manifold without boundary. The $\mathcal{F}$-functional
is given as

(1.1) $\mathcal{F}(g,u)=\int_{M}(4|\nabla u|^{2}+R_{g}u^{2})d{\rm vol}_{g},$

where $R_{g}$ is the scalar curvature of $g$ and $u>0$ is a smooth function.
Perelman’s $\lambda$-functional is then defined as

(1.2) $\lambda(g)=\inf\left\\{\mathcal{F}(g,u)\mid\int_{M}u^{2}d{\rm
vol}_{g}=1\right\\}.$

Clearly, from $(\ref{eqn: lambda-functional})$ and $(\ref{eqn:
F-functional2})$, $\lambda(g)$ is the smallest eigenvalue of the Schrödinger
operator $-4\Delta_{g}+R_{g}$. Starting from this spectral viewpoint, we
developed Perelman’s $\lambda$-functional on compact manifolds with isolated
conical singularities in [DW18].

To take the scale into account, Perelman also introduced the $W$-functional
and $\mu$-functional on smooth compact manifolds in [Per02]. They play
important roles in the study of singularities of the Ricci flow. The
$W$-functional is given by

$W(g,u,\tau)=\frac{1}{(4\pi\tau)^{\frac{n}{2}}}\int_{M}[\tau(R_{g}u^{2}+4|\nabla
u|^{2})-2u^{2}\ln u-nu^{2}]d{\rm vol}_{g},$

where $\tau>0$ is a scale parameter, and $u>0$ is a smooth function. The
$\mu$-functional is defined by, for each $\tau>0$,

(1.3) $\mu(g,\tau)=\inf\left\\{W(g,u,\tau)\mid u\in C^{\infty}(M),\ \ u>0,\ \
\frac{1}{(4\pi\tau)^{\frac{n}{2}}}\int_{M}u^{2}d{\rm vol}_{g}=1\right\\}.$

It is well-known that the finiteness of the infimum in $(\ref{eqn: mu-
functional})$ follows from the logarithmic Sobolev inequality on smooth
compact Riemannian manifolds, while the regularity of the minimizer follows
from the elliptic estimates and Sobolev embedding.

In view of the applications to Ricci flow, it is a natural question to develop
these functionals for singular spaces. We refer to the excellent survey
[Bam21] for some recent developments in Ricci flow and the role played by
singular spaces. The possibility of conical singularity appearing in type I
singularities of Ricci flow is indicated by the recent work [MW17]. The work
[GS18] studies the Ricci flow coming out of manifolds with isolated conical
singularities. Conical singularity can also be viewed as a generalization of
orbifold singularity and we refer to [KL14] for Ricci flow on orbifolds and
its applications.

In [DW20], based on the work in [DW18], we studied the $W$-functional and
$\mu$-functional on manifolds with isolated conical singularities. In [Ozu20],
Ozuch developed Perelman’s functionals on manifolds with isolated conical
singularities from a different point of view. In [KV21], Kröncke and Vertman
obtained a partial asymptotic expansion for functions $u$, satisfying
$(\Delta_{g}+cR_{g})u=F(u)$ with $F(u)\leq C|u\ln u|$, near isolated conical
singularities whose cross sections have scalar curvature the same as the round
sphere, and further studied singular Ricci solitons in this case, see Remark
1.7 for more details.

Non-isolated conical singularity plays a crucial role in the resolution of the
Yau-Tian-Donaldson conjecture [CDS15, T15]; see also [S18, LZ17] for some
related work on Ricci flow. In this paper, we develop Perelman’s functionals
on compact manifolds with non-isolated conical singularities as defined in
Definition 2.1 below, and also on manifolds with $r^{\alpha}$-horn
singularities as defined in Definition 7.1 below. Our definition of non-
isolated singularities includes isolated singularities. In particular, we
establish the existence of the minimizers and obtain asymptotic behavior of
the minimizers near the singularity. These asymptotic behaviors are not
optimal for non-isolated conical singularity. On the other hand, we obtain an
optimal partial asymptotic expansion for eigenfunctions of $-\Delta+cR$ near a
general isolated conical singularity.

###  1.1. Spectral properties of the operator $-4\Delta+R$ on manifolds with
non-isolated conical singularities

Roughly speaking, a compact manifold with non-isolated conical singularities
$(M^{n},g)$ is a compact manifold $M$ with boundary $\partial M$ and a
Riemannian metric $g$ degenerating in a specific way at the boundary. The
boundary $\partial M$ is the total space of the fibration $\pi:\partial
M\rightarrow B$ with the $f$-dimensional compact manifold $F^{f}$ as a typical
fiber, and in a collar neighborhood $(0,1)\times\partial M$ of $\partial M$
the metric $g$ is asymptotic to $g_{0}=dr^{2}+r^{2}\hat{g}+\pi^{*}\check{g}$,
where $r$ is a coordinate on $(0,1)$, $\check{g}$ is a metric on the base $B$
and $\hat{g}$ is the restricted metric on fibers $F$. These singularities are
also called wedge singularities (e.g. [Maz91]) or edge singularities (e.g.
[Ver21]). Vertman [Ver21] established the existence of Ricci de Turk flow
within the class of these singular manifolds whose curvature satisfy some
asymptotic control near singularities. Note that when $B$ is a single point,
we are reduced to manifolds with isolated conical singularity.

Our first main result is the following spectral property of the operator
$-4\Delta+R$ on these singular manifolds. This enables us to define Perelman’s
$\lambda$-functional on these singular manifolds as the first eigenvalue of
$-4\Delta+R$.

######  Theorem 1.1 (Theorem 4.4 below).

Let $(M^{n},g)$, $n\geq 4$, be a compact Riemannian manifold with non-isolated
conical singularities as defined in Definition 2.1 below. If
$\min\limits_{\partial M}\\{R_{\hat{g}}\\}>(f-1)$, then the operator
$-4\Delta_{g}+R_{g}$ with domain $C^{\infty}_{0}(\mathring{M})$ is semi-
bounded, and the spectrum of its Friedrichs extension consists of discrete
eigenvalues with finite multiplicities
$\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\cdots,$ and
$\lambda_{k}\rightarrow+\infty$, as $k\rightarrow+\infty$. The corresponding
eigenfunctions form a complete basis of $L^{2}(M)$

######  Remark 1.2.

Similarly Theorem 1.1 extends to the Schrödinger operator $-\Delta_{g}+cR_{g}$
for any constant $c$, where the scalar curvature condition becomes
$\min\limits_{\partial M}\\{R_{\hat{g}}\\}>\frac{f[4c(f-1)-(f-2)]-1}{4c}$. For
example, for the well known conformal Laplacian
$-\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}$, the scalar curvature condition becomes
$\min\limits_{\partial M}\\{R_{\hat{g}}\\}>\frac{(f-1)(n-f-1)}{n-2}$.

######  Remark 1.3.

The role of the Schrödinger operator $-\Delta_{g}+\frac{1}{2}R_{g}$ in the
study of scalar curvature has attracted a lot attention recently [Sch17]. In
[Gro21], Gromov further highlighted the importance of the operator
$-\Delta_{g}+\frac{1}{2}R_{g}$ in the study of scalar curvature. In
particular, he asks for what class of isolated conical singularity this
operator will be positive. Our result gives an answer to this question for
general conical singularity. Namely, for non-isolated conical singularity, a
sufficient condition is

$\min\limits_{\partial M}\\{R_{\hat{g}}\\}>\frac{1}{2}(f^{2}-1).$

For isolated conical singularity, this becomes

$\min\limits_{\partial M}\\{R_{\hat{g}}\\}>\frac{1}{2}(n-2)n.$

Moreover, we obtain the following asymptotic behavior for eigenfuctions of
$-4\Delta_{g}+R_{g}$ near the singular set $\partial M$, by a Nash-Moser
iteration argument.

######  Theorem 1.4 (Corollary 5.2 below).

Let $u$ be an eigenfunction of $-4\Delta_{g}+R_{g}$ on a $n$-dimensional
manifold $(M^{n},g)$ with non-isolated conical singularities. Then

(1.4) $|\nabla^{i}u|=o(r^{-\frac{n-2}{2}-i})$

as $r\rightarrow 0$, i.e. approaching the singular set $\partial M$, for
$i=0,1$. Here $r$ is the radial variable on a conical neighborhood of the
singular set $\partial M$.

######  Remark 1.5.

(1.4), as well as (1.6) below, is in general not optimal as one expects that
the exponent $-\frac{n-2}{2}-i$ should be replaced by $-\frac{f-2}{2}-i$.
Later, we will establish an optimal partial asymptotic expansion for
eigenfunctions of $-\Delta+cR$ for isolated conical singularity.

###  1.2. The $W$-functional on manifolds with non-isolated conical
singularities

For the $W$-functional, we extend the results in [DW20] and obtain:

######  Theorem 1.6.

Let $(M^{n},g)$ be a compact manifold with non-isolated conical singularities.
If $\min\limits_{\partial M}\\{R_{\hat{g}}\\}>(f-1)$, then for each fixed
$\tau>0$,

(1.5) $\mu(g,\tau)=\inf\left\\{W(g,u,\tau)\mid u\in
W^{1,2}_{1-\frac{n}{2}}(M),u>0,\left\|\frac{1}{(4\pi\tau)^{\frac{n}{2}}}u\right\|_{L^{2}(M)}=1\right\\}>-\infty.$

Here $W^{1,2}_{1-\frac{n}{2}}(M)$ is a weighted Sobolev space defined in
$(\ref{eqn: WSN})$ below.

Moreover, there exists a smooth function $u>0$ that realizes the infimum in
$(\ref{eqn: finite-infimum})$. The minimizer satisfies

(1.6) $|\nabla^{i}u|=o(r^{-\frac{n-2}{2}-i}),$

as $r\rightarrow 0$, for $i=0,1$. Here $r$ is the radial variable of a conical
neighborhood of the singular set.

We would like to mention that Theorem 1.6 extends to the expander entropy
$W_{+}$-functional introduced in [FIN05]. In this paper we will only focus on
$W$-functional, since the discussion for $W_{+}$-functional is the same.

The finiteness of infimums of $W$-functional and $W_{+}$-functional, and the
existence of the minimizer follow from the direct method in the calculus of
variations, with the help of the (weighted and unweighted) Sobolev embeddings
and compactness of some weighted Sobolev embeddings on manifolds with non-
isolated conical singularities obtained in §3. The asymptotic behavior of the
minimizing function then follows from a Nash-Moser iteration argument as
discussed in §6.

In contrast to isolated conical singularities, the scaling technique, which is
used in [DW18], does not work any more for deriving weighted Sobolev
inequalities and weighted elliptic estimate on manifolds with non-isolated
conical singularities. Instead, we adapt the partition argument for an
isolated cone used in [DW20] to the non-isolated conical singularities
setting, and obtain the (unweighted) Sobolev inequality. A Hardy inequality
plays a critical role in this derivation, as well as in the semi-boundedness
estimate for the operator $-4\Delta+R$. Then we obtain the weighted Sobolev
inequality from the Sobolev inequality.

One can further analyze the asymptotic behavior of the functional
$\mu(g,\tau)$ as $\tau\rightarrow+\infty$ and $\tau\rightarrow 0$ to obtain
the finiteness for the $\nu$-entropy on manifolds with non-isolated conical
singularities. Indeed, from the Sobolev inequality in Proposition 3.1, the
logarithmic Sobolev inequality follows. Then, as in the smooth case (see, e.g.
Lemma 6.30 in [CCG+07]), one deduces from the logarithmic Sobolev inequality
that, when $\min\limits_{\partial M}\\{R_{\hat{g}}\\}>f-2$,

(1.7) $\mu(g,\tau)\rightarrow\left\\{\begin{array}[]{cc}+\infty,\ \ \mbox{if}\
\lambda(g)>0,\\\ -\infty,\ \ \mbox{if}\ \lambda(g)<0.\end{array}\right.$

On the other hand, one can show the finiteness of the limit
$\lim\limits_{\tau\rightarrow 0}\mu(g,\tau)$ on manifolds with non-isolated
conical singularities, by combining the scaling invariance property:
$\mu(cg,c\tau)=\mu(g,\tau)$ for any $c>0$, with the fact that
$\mu(g_{\mathbb{R}^{n}},1)=0$, as well as $\mu(g_{C(F)},1)<\infty$. Here
$g_{\mathbb{R}^{n}}$ is the Euclidean metric on $\mathbb{R}^{n}$, and
$g_{C(F)}$ is the model cone metric on $(0,+\infty)\times F$. The finiteness
of $\mu(g_{C(F)},1)$ is due to Ozuch [Ozu20]. Therefore, we obtain that

(1.8) $\nu(g)=\inf\\{\mu(g,\tau)\mid\tau>0\\}>-\infty,$

on manifolds with non-isolated conical singularities $(M,g)$ satisfying
$\min_{\partial M}\left\\{R_{\hat{g}}\right\\}>f-2$, provided that
$\lambda(g)>0$.

The finiteness of $\nu$-entropy and Theorems 1.1 and 1.6, as announced in
[DW20], answer the existence problem of Perelman’s functionals on manifolds
with non-isolated (edge) conical singularities, which was also raised in the
survey [KV20].

###  1.3. The horn singularity

We also study the spectrum of Schrödinger operator $-\Delta+cR$ and Perelman’s
functionals on manifolds with (either non-isolated or isolated) more general
singularities, i.e. $r^{\alpha}$-horn in [Che80], whose model is $((0,1)\times
F^{f},g_{\alpha}=dr^{2}+r^{2\alpha}\hat{g})$ with $\alpha\in\mathbb{N}$ and
$\hat{g}$ a Riemannian metric on $F^{f}$. This type of singularity occurs
naturally in general relativity (e.g., the negative mass Schwarzschild metric
has a horn singularity [ST18]) and singular projective variety [HP85].

The study of analysis on manifolds with $r^{\alpha}$-horn singularities with
$\alpha>1$ is very different from the case of conical singularities (i.e. the
case of $\alpha=1$). For example, neither Hardy inequality nor the partition
method used in [DW20] and in the proof of Proposition 3.1 below works for
$r^{\alpha}$-horn singularities with $\alpha>1$. However, we still establish
some $\alpha$-weighted Sobolev inequalities in Proposition 7.3 below as well
as some compact $\alpha$-weighted Sobolev embeddings in Proposition 7.4 below,
by using some change of variables to relate the problem to that on cylinders.
Then we obtain:

######  Theorem 1.7.

Let $(M^{n},g)$ be a compact Riemannian manifold with $r^{\alpha}$-horn
singularities as in Definition 7.1 below. If $\min\limits_{\partial
M}\\{R_{\hat{g}}\\}>0$, then

  1. (1)

the Friedrichs extension of the Schrödinger operator $-\Delta_{g}+cR_{g}\
(c>0)$ with the domain $C^{\infty}_{0}(\mathring{M})$ has the spectrum
consisting of discrete eigenvalues with finite multiplicity
$-\infty<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\cdots$, and the
corresponding eigenfunctions are smooth and form a basis of $L^{2}(M,g)$,

  2. (2)

$\inf\left\\{W(g,\tau,u)\mid u\in
W^{1,2}_{\alpha-\frac{n\alpha}{2},\alpha}(M),\ \
\|u\|_{L^{2}(M)}=(4\pi\tau)^{\frac{n}{2}}\right\\}>-\infty,$

  3. (3)

and an eigenfuction of $-\Delta+cR$ or the minimizer of the $W$-functional,
$u$, has the asymptotic behavior

$|\nabla^{i}u|=o(r^{-\frac{n-2}{2}\alpha-i\alpha})$

as $r\rightarrow 0$, i.e. approaching the singular set, for $i=0,1$. Here $r$
is the radial variable of a horn-like neighborhood of the singular set.

######  Remark 1.8.

Note that the assumption for scalar curvature on the cross section in Theorem
1.7 is weaker than that in the conical singularity case, Cf. Theorems 1.1 and
1.6, and is independent of the dimension of the manifold and the constant $c$
in the Schrödinger operator.

###  1.4. A partial asymptotic expansion of eigenfunction on manifolds with
isolated conical singularities

As we mentioned, the asymptotic behaviors of the minimizers, important in
application, are not optimal. This can be much improved for isolated conical
singularities. Indeed we obtain the following optimal partial asymptotic
expansion for the eigenfunctions, by using Melrose’s b-calculus theory [Mel93]
and a weighted elliptic bootstrapping argument.

######  Theorem 1.9 (Theorem 8.3 below).

Let $u$ be an eigenfunciton of $-4\Delta_{g}+R_{g}$ on a $n$-dimensional
manifold with an isolated conical singularity. Then

(1.9) $u=a(x)r^{\mu}+o(r^{\mu^{\prime}})$

as $r\rightarrow 0$, where $\mu=-\frac{n-2}{2}+\frac{\sqrt{\nu-(n-2)}}{2}$ and
$a(x)$ is an eigenfunction with eigenvalue $\nu$ of
$-4\Delta_{\hat{g}}+R_{\hat{g}}$ on the cross section $(F,\hat{g})$ of the
cone, and $\mu^{\prime}>\mu$. Here $r$ is the radial variable on the conical
neighborhood of the singular point.

This in particular implies the partial asymptotic expansion for eigenfunctions
obtained by Kröncke and Vertman in [KV21], where they considered the case
where $(F,\hat{g})$ has the scalar curvature $R_{\hat{g}}=(n-1)(n-2)$, i.e.,
that of the round sphere. In this special case, one can easily see that
$\mu\geq 0$ in Theorem 1.9.

On manifolds with non-isolated conical singularities with some additional
assumptions, e.g. the metrics on the fibres are isospectral with respect to
the operator $-4\Delta+4R$ and the difference between asymptotic conical
metric and model conical metric is smooth up to the singular set, the work of
Mazzeo in [Maz91] gives a full asymptotic expansion for the eigenfunctions.
Note that these restrictions are not imposed in our work for manifolds with
non-isolated conical singularities.

###  1.5. Gradient Ricci solitons with isolated conical singularities

Gradient Ricci solitons are singularity models of Ricci flow. We show that
there are no non-trivial gradient steady or expanding Ricci soliton on compact
manifolds with isolated conical singularity, provided the scalar curvature of
the cross sections of the conical singularities equals that of the sphere. For
this purpose, we first obtain, with the help of Melrose’s b-calculus, some
asymptotic estimate as in Proposition 9.3 below for the potential functions of
compact gradient Ricci solitons with isolated conical singularities. In
particular, we show that the potential function of a compact gradient Ricci
soliton with isolated conical singularities is bounded and goes to zero when
one approaches the singular points, when the above condition on the scalar
curvature is satisfied. Using this, if one further assumes that the soliton is
either steady or expanding, then we show that it must be Einstein. Here the
subtlety is that we do not impose asymptotic condition for the potential
functions, and thus, a priori, gradient Ricci solitons with conical
singularities may not be critical points of Perelman’s functionals, unlike the
smooth case.

######  Theorem 1.10.

Let $(M^{n},g,f)$ be a compact steady or expanding gradient Ricci soliton with
isolated conical singularities, i.e.

(1.10) $Ric_{g}+\nabla^{2}f=\Lambda g,$

with constant $\Lambda\leq 0$. If the cross section of the model cone at each
singularity has scalar curvature $R_{\hat{g}}=(n-1)(n-2)$, then $(M^{n},g)$
must be an Einstein manifold with isolated conical singularities, i.e.
$Ric_{g}=\Lambda g.$

######  Remark 1.11.

More generally, Kröncke and Vertman [KV21] have proved that compact steady and
expanding Ricci solitons with isolated conical singularities and
dimension$\geq 4$ have to be Einstein, provided that the scalar curvature of
cross section of model cones are the same as that of the round sphere. In
their argument, in the 4 dimensional case, some asymptotic control for Ricci
curvature near the singularities was assumed. Our results hold with neither
dimensional restriction nor any extra condition other than the assumption on
the scalar curvature.

###  1.6. Organization of the paper

The article is organized as follows. In §2, we give the precise definition for
manifolds with non-isolated conical singularities and recall some basic facts
for Riemannian submersions. In §3, we derive the Sobolev inequality and some
weighted Sobolev inequalities on manifolds with non-isolated conical
singularities. We also derive weighted elliptic regularity estimate on
manifolds with non-isolated conical singularities. In §4, we establish the
desired spectral property for the Schrödinger operator $-4\Delta+R$ on
manifolds with non-isolated conical singularities. In §5 and 6, we obtain
asymptotic behavior for eigenfunctions of $-4\Delta+R$ and the minimizer of
the $W$-functional by using Nash-Moser iteration. In §7, we study the spectrum
of $-\Delta+cR$ and Perelman’s functionals on manifolds with $r^{\alpha}$-horn
singularities. In §8, we derive a partial asymptotic expansion for
eigenfunctions of $-4\Delta+R$ on manifolds with isolated conical
singularities by using the b-calculus theory of Melrose and a weighted
elliptic bootstrapping argument. In §9, as another application of b-calculus
theory and the asymptotic estimates for functions in weighted Sobolev spaces,
we prove that steady and expanding gradient Ricci solitons with isolated
conical singularities must be Einstein, provided that the scalar curvature of
cross section of model cones are the same as that of the standard sphere.

Acknowledgements: The authors would like to thank an anonymous referee whose
valuable comments and suggestions have been very helpful in improving the
presentation of this work.

##  2\. Manifolds with non-isolated conical singularities

######  Definition 2.1.

We say that $(M^{n},g)$ is a compact Riemannian manifold with non-isolated
conical singularities, if

  1. (1)

$M^{n}$ is a compact $n$-dimensional manifold with boundary $\partial M$.

  2. (2)

The boundary $\partial M$ endowed with a Riemannian metric $g_{\partial M}$ is
the total space of a Riemannian submersion

(2.1) $\pi:(\partial M,g_{\partial M})\longrightarrow(B^{b},\check{g})$

with a $b$-dimensional closed Riemannian manifold $(B^{b},\check{g})$ as the
base and a $f$-dimensional closed manifold $F^{f}$ as a typical fiber; in
particular, $b+f=n-1$, and

(2.2) $g_{\partial M}=\hat{g}+\pi^{*}\check{g},$

where $\hat{g}$ denotes metrics on fibers induced by $g_{\partial M}$.

  3. (3)

Moreover, there exists a collar neighborhood, $U:=(0,1)\times\partial M,$ of
the boundary $\partial M$ in $M$ with the radial coordinate $r$ $(r=0$
corresponds to the boundary $\partial M)$, and the metric $g$ restricted on
$U$ can be written as $g=g_{0}+h$, where the Riemannian metric $g_{0}$ on $U$
is given by

(2.3) $g_{0}=dr^{2}+r^{2}\hat{g}+\pi^{*}\check{g},$

and 2-tensor $h$ satisfies

(2.4) $|r^{k}\nabla^{k}_{g_{0}}h|_{g_{0}}=O(r^{\alpha}),\quad\text{as}\ \
r\rightarrow 0,$

for some $\alpha>0$ and all $k\in\mathbb{N}$.

######  Remark 2.2.

We do not require the high order term $h$ in Definition 2.1 to be smooth up to
singularities.

######  Remark 2.3.

Note that the Riemannian metric $g_{\partial M}$ on $\partial M$ is not the
usual restriction of the Riemannian metric $g$ to the boundary. In fact, in
metric sense, the boundary $\partial M$ is collapsed fiberwisely to the base
$B$. Thus $M$ is metrically completed by adding $B$ which forms the non-
isolated singularities of the completed space. Along the normal direction of
each point of this singular stratum, the metric is asymptotic to a cone over a
typical fiber.

Now we briefly recall some basic facts about Riemannian submersions, and for
more details we refer to [ONe66] and Chapter 9 in [Bes87]. We consider the
Riemannian submersion given in (2.1). Let $\mathcal{H}$ and $\mathcal{V}$
denote respectively the horizontal and vertical projections. Let
$\nabla^{\partial M}$ denote the Levi-Civita connections on $(\partial
M,g_{\partial M})$. The O’Neill’s $A$-tensor and $T$-tensor are given by:

$\displaystyle A_{X}Y$ $\displaystyle=\mathcal{H}\nabla^{\partial
M}_{\mathcal{H}X}\mathcal{V}Y+\mathcal{V}\nabla^{\partial
M}_{\mathcal{H}X}\mathcal{H}Y,$ $\displaystyle T_{X}Y$
$\displaystyle=\mathcal{H}\nabla^{\partial
M}_{\mathcal{V}X}\mathcal{V}Y+\mathcal{V}\nabla^{\partial
M}_{\mathcal{V}X}\mathcal{H}Y,$

for any vector fields $X$ and $Y$ on $\partial M$. The tensor $T$ is related
to the second fundamental form of fibers, and it vanishes identically if and
only if each fiber is totally geodesic.

By the formula (9.37) in [Bes87], the scalar curvature of $g_{0}$ in
$(\ref{eqn: model conical metric})$ is given by

(2.5)
$R_{g_{0}}=\frac{1}{r^{2}}(R_{\hat{g}}-f(f-1))+R_{\check{g}}\circ\pi-|A|^{2}-|T|^{2}-|N|^{2}-\check{\delta}N,$

where the horizontal vector field $N=\sum\limits^{f}_{j=1}T_{V_{j}}V_{j}$ is
the mean curvature vector of fibers,
$\check{\delta}N=-\sum\limits^{b}_{i=1}g_{\partial M}(\nabla^{\partial
M}_{X_{i}}N,X_{i})$, and $\\{V_{1},\cdots,V_{f},X_{1},\cdots,X_{b}\\}$ is a
local orthonormal frame for $T(\partial M)$ with respect to $g_{\partial
M}=\hat{g}+\pi^{*}\check{g}$ such that $V_{1},\cdots,V_{f}$ are vertical and
$X_{1},\cdots,X_{b}$ are projectable and horizontal.

##  3\. Sobolev and Weighted Sobolev spaces

In this section, we recall some weighted Sobolev spaces on compact Riemannian
manifolds with non-isolated conical singularities. We also review and
establish some (unweighted) Sobolev and weighted Sobolev embeddings on these
singular manifolds that we will use in the following sections.

Various weighted Sobolev spaces have been introduced and studied on manifolds
with conical singularities in [Maz91], [BP03], [CLW12], [Beh13], [DW18],
[DW20], [Oik20], [KV21] and [Ver21]. These weighted Sobolev spaces turn out to
be very useful in the study of elliptic and heat equation problems on
conically singular manifolds. In particular, weighted Sobolev inequality and
weighted elliptic regularity estimate are important tools in these studies.

Besides weighted Sobolev inequality, in the study of $W$-functional the
(unweighted) Sobolev inequality is also necessary (see, §6 below, and [DW20]).
In [DY], the usual $L^{2}$-Sobolev inequality on compact Riemannian manifolds
with isolated conical singularities has been established by choosing a
suitable partition for a cone and applying a Hardy inequality. We then noticed
that their idea also works for $L^{p}$-Sobolev inequality for $1<p<n$ in
[DW20]. Later, Oikonomopoulos [Oik20] employed this idea to study the
$C^{\infty}_{0}$ density problem for weighted Sobolev spaces on manifolds with
conical singularities. Now we further adapt this idea to non-isolated conical
singularities case, and obtain the Sobolev inequality in Proposition 3.1. To
our knowledge, the (unweighted) Sobolev inequality on manifolds with non-
isolated conical singularities and its proof are not in the literature. So we
provide that in §3.1.

In the isolated conical singularities case, thanks to the homogeneity of cone
metric along the radial direction, weighted Sobolev inequality and elliptic
estimate were obtained in [Beh13] (also see [DW18]) by employing the scaling
technique, which had been used by Bartnik [Bar86] for asymptotically Euclidean
manifolds. However, in the non-isolated conical singularities case, due to the
presence of $\pi^{*}\check{g}$ term, the model singular metric $g_{0}$ in
$(\ref{eqn: model conical metric})$ does not have such homogeneity, and as a
result, the scaling technique does not work any more. Instead, in §3.2, we
derive the weighted Sobolev inequality in the non-isolated conical
singularities case from the (unweighted) Sobolev inequality, which is obtained
in Proposition 3.1. The local version of the weighted Sobolev inequality was
obtained in [CLW12] in a different way.

In §3.3, we then provide two compact weighted Sobolev embeddings in
Propositions 3.8 and 3.9 that are used in the proof of Theorem 4.4 and in the
study of the $W$-functional in §6. These compact embedding results are
generalizations, to the non-isolated conical singularities, of our previous
results in Theorem 3.1 in [DW18] and Proposition 3.6 in [DW20] for isolated
conical singularities.

In §3.4, we establish a weighted elliptic estimate in Proposition 3.10. Again
because the lack of the homogeneity of the model singular metric $g_{0}$ in
$(\ref{eqn: model conical metric})$, we are not able to prove a weighed
elliptic estimate in the non-isolated conical singularities case by using
scaling technique, which was used in the isolated conical singularities case
in [Beh13] (also see [DW18]). We notice that a weighted elliptic estimate on
manifolds with (either isolated or non-isolated) conical singularities can be
proved by doing integration by parts, using the asymptotic control of
Riemannian curvature tensor near the conical singularities, and an
approximation argument. This idea also works for compact smooth manifolds, and
should also work for various non-compact manifolds with certain asymptotic
controls for the metric tensor. As this proof is not in the literature, we
provide it in §3.4.

###  3.1. Sobolev spaces and embedding

For any $k\in\mathbb{N}$ and $p\geq 1$, as usual, the Sobolev space
$W^{k,p}(M)$ is the completion of $C^{\infty}_{0}(\mathring{M})$ with respect
to the Sobolev norm

(3.1)
$\|u\|_{W^{k,p}(M)}=\left(\int_{M}\left(\sum^{k}_{i=0}|\nabla^{i}u|^{p}d{\rm
vol}_{g}\right)\right)^{\frac{1}{p}},$

where $\mathring{M}:=M\setminus\partial M$ is the interior of $M$.

######  Proposition 3.1.

Let $(M^{n},g)$ be a compact Riemannian manifold with non-isolated conical
singularities as in Definition 2.1.

  1. (1)

For each $1<p<n$ with $\frac{np}{n-ip}\neq f$, $i=0,\cdots,k-l-1$, one has

(3.2) $\|u\|_{W^{l,q}(M)}\leq C(M,g,p,k)\|u\|_{W^{k,p}(M)},$

for any $u\in C^{\infty}_{0}(\mathring{M})$ and any $1\leq q\leq q_{l}$, where
$C(M,g,p,k)$ is a constant, and $l<k$ and $q_{l}$ satisfy
$\frac{1}{q_{l}}=\frac{1}{p}-\frac{k-l}{n}>0$. Hence, there is a continuous
embedding $W^{k,p}(M)\subset W^{l,q}(M)$, for any $1\leq q\leq q_{l}$.

  2. (2)

For $k,j\in\mathbb{N}$ with $k-j>\frac{n}{p}$, one has

(3.3) $\|u\|_{C^{j}(M)}\leq C(M,g,p,k)\|u\|_{W^{k,p}(M)},$

for any $u\in C^{\infty}_{0}(\mathring{M})$. Hence, there is a continuous
embedding $W^{k,p}(M)\subset C^{j}(M)$. Here $C^{j}(M)$ is the completion of
$C^{\infty}_{0}(\mathring{M})$ with respect to the norm

$\|u\|_{C^{j}(M)}:=\max\limits_{0\leq l\leq j}\sup\limits_{M}|\nabla^{l}u|.$

  3. (3)

If $k,l,j\in\mathbb{N}$ satisfy $k-j-l>\frac{n}{p}$, then any $u\in
W^{k,p}(M)$ satisfies

(3.4) $|\nabla^{j}u|\leq C(M,g,k,l,j,)\|u\|_{W^{k,p}(M)}r^{l},$

near the singular set $r=0$.

Clearly, the key in the derivation of the above Sobolev inequalities is to
establish them on the collar neighborhood $U$ of $\partial M$ in the item (3)
in Definition 2.1. As in [DY] and [DW20], a Hardy inequality (see, e.g. 330 on
p. 245 in [HLP34]) will play a crucial rule. So we recall it: for any $p>1$
and $a\neq 1$, one has

(3.5)
$\int^{\infty}_{0}|u|^{p}x^{-a}dx\leq\left(\frac{p}{|a-1|}\right)^{p}\int^{\infty}_{0}|u^{\prime}(x)|^{p}x^{p-a}dx,$

for any $u\in C^{\infty}_{0}((0,\infty))$. This then implies a Hardy
inequality (c.f. (3.8) in [DW20]) on $U:=(0,1)\times\partial M$ with the
metric $g_{0}=dr^{2}+r^{2}\hat{g}+\pi^{*}\check{g}$: for $p>1$ and
$k\in\mathbb{N}$ with $pk\neq f$, and any $u\in C^{\infty}_{0}(U)$, we have

(3.6) $\displaystyle\int_{U}\frac{|u|^{p}}{r^{pk}}d{\rm vol}_{g_{0}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{p}{|f-pk|}\right)^{p}\int_{U}\frac{|\nabla
u|^{p}_{g_{0}}}{r^{p(k-1)}}d{\rm vol}_{g_{0}}.$

######  Lemma 3.2.

For $1<p<n$ with $p\neq f$, there is a constant $C$ depending only on $g_{0}$
and $p$ such that

(3.7) $\|u\|_{L^{q}(U,g_{0})}\leq C\|\nabla u\|_{L^{p}(U,g_{0})},$

for any $u\in C^{\infty}_{0}(U)$, where $q=\frac{np}{n-p}$.

For $p>n$, there is a constant $C$ depending only on $g_{0}$ and $p$ such that

(3.8) $\|u\|_{C^{0}(U)}\leq C\|u\|_{W^{1,p}(U)}$

for any $u\in C^{\infty}_{0}(U)$.

###### Proof.

First of all, we note that because $B$ is compact, the metrics restricted to
fibers $\hat{g}|_{\pi^{-1}(y)}$ are uniformly quasi-isometric, i.e. there
exists a constant $C$ and a Riemannian metric $\hat{g}_{0}$ on $F^{f}$ such
that $\frac{1}{C}\hat{g}_{0}\leq\hat{g}|_{\pi^{-1}(y)}\leq C\hat{g}_{0}$ for
all $y\in B$.

Now we choose a finite open cover of the collar neighborhood $U$ as follows,
so that on each piece the metric $g_{0}$ is quasi-isometric to the standard
Euclidean metric on $\mathbb{R}^{n}$.

First, choose a finite open cover $\\{U_{i}\\}$ of B, such that each
$\\{U_{i},\check{g}|_{U_{i}}\\}$ is quasi-isometric to an open domain in the
standard Euclidean space $\mathbb{R}^{b}$, and the submersion $\pi:\partial
M\rightarrow B$ can be locally trivialized over this open cover, i.e. there
exists diffeomorphisms $\varphi_{i}:\pi^{-1}(U_{i})\rightarrow U_{i}\times F$.
Let $\check{\rho}_{i}$ be a partition of unity subordinate to this open cover
of $B$.

Next, choose a finite open cover $\\{V_{j}\\}$ of $F$, such that each $V_{j}$
can be embedded into Euclidean unite sphere $\mathbb{S}^{f}$ and the metric
$\hat{g}_{0}$ restricted to each $V_{j}$ is quasi-isometric to the the round
sphere metric. Let $\hat{\rho}_{j}$ be a partition of unity subordinate to
this open cover of $F$.

So $\\{\varphi^{-1}_{i}(U_{i}\times V_{j})\\}$ form a finite open cover of
$\partial M$, and further we obtain a finite open cover
$\\{(0,1)\times\varphi^{-1}_{i}(U_{i}\times V_{j})\\}$ of
$U=(0,1)\times\partial M$, with a subordinated partition of unity
$\rho_{ij}:=\tilde{\pi}^{*}\big{(}[\pi^{*}(\check{\rho_{i}})]\cdot[(({\rm
Pr}_{2})\circ\varphi_{i})^{*}(\hat{\rho}_{j})]\big{)}$, where
$\tilde{\pi}:(0,1)\times\partial M\rightarrow\partial M$ and ${\rm
Pr}_{2}:U_{i}\times F\rightarrow F$ are natural projections. Then by the
construction, one easily sees that $g_{0}$ restricted to
$(0,1)\times\varphi^{-1}_{i}(U_{i}\times V_{j})$ is quasi-isometric to the
standard Euclidean metric on $\mathbb{R}^{n}$, and more importantly

(3.9) $|d\rho_{ij}|_{g_{0}}(r,\theta)\leq C_{ij}r^{-1},$

where $C_{ij}$ are some constants.

The usual Sobolev inequality on standard Euclidean spaces together with the
estimate (3.9) and the Hardy inequality (3.6) imply the Sobolev inequality
(3.7) on the collar neighborhood $U$ with metric
$g_{0}=dr^{2}+r^{2}\hat{g}+\pi^{*}\check{g}$.

For Morrey’s inequality (3.8), let $u\in C^{\infty}_{0}(U)$, because
$u=\sum\limits_{i,j}\rho_{ij}u$, one has

$\displaystyle\|u\|_{C^{0}(U)}$ $\displaystyle\leq$
$\displaystyle\sum_{i,j}\|\rho_{ij}u\|_{C^{0}(U)}$ $\displaystyle\leq$
$\displaystyle\sum_{i,j}C\|\rho_{ij}u\|_{W^{1,p}(U)}\qquad\qquad(\text{Morrey's
inequality on}\ \ \mathbb{R}^{n})$ $\displaystyle\leq$
$\displaystyle\sum_{i,j}C\left(\int_{U}\left(|d(\rho_{ij}u)|^{p}_{g_{0}}+|\rho_{ij}u|^{p}\right)d{\rm
vol}_{g_{0}}\right)^{\frac{1}{p}}$ $\displaystyle\leq$
$\displaystyle\sum_{i,j}C\left(\int_{U}\left(|d\rho_{ij}|^{p}_{g_{0}}|u|^{p}+|\rho_{ij}|^{p}|du|^{p}_{g_{0}}+|\rho_{ij}u|^{p}\right)d{\rm
vol}_{g_{0}}\right)^{\frac{1}{p}}$ $\displaystyle\leq$ $\displaystyle
C\left(\int_{U}\left(r^{-p}|u|^{p}+|du|^{p}_{g_{0}}+|u|^{p}\right)d{\rm
vol}_{g_{0}}\right)^{\frac{1}{p}}\quad(\text{Estimate}(\ref{eqn: cone-
partition-of-unity-derivative}))$ $\displaystyle\leq$ $\displaystyle
C\left(\int_{U}\left(|du|^{p}_{g_{0}}+|u|^{p}\right)d{\rm
vol}_{g_{0}}\right)^{\frac{1}{p}}\qquad(\text{Hardy inequality }(\ref{eqn:
cone-Hardy-inequality}))$ $\displaystyle=$ $\displaystyle
C\|u\|_{W^{1,p}(U)}.$

In the above derivation, the constant $C$ may vary along steps. ∎

Proof of Proposition 3.1. (1) By using the usual Sobolev inequalities on
compact manifolds and applying the Kato’s inequality:
$|\nabla|\nabla^{k}u||\leq|\nabla^{k+1}u|$, Lemma 3.2 implies the Sobolev
inequality in (3.2).

(2) By combining the inequality (3.8) and Morrey’s inequality on compact
smooth manifolds, one has that for any $q>n$ there is a constant $C$ such that

(3.10) $\|u\|_{C^{0}(M)}\leq C\|u\|_{W^{1,q}(M)}$

for any $u\in W^{1,q}(M)$. As in the smooth case, the general inequality in
$(\ref{eqn: Sobolev Cr inequality})$ then follows from the Sobolev inequality
$(\ref{eqn: Sobolev inequality})$ and the Morrey’s inequality $(\ref{eqn:
Morrey inequality})$: see e.g. §5.6.3 in [Eva] for more details.

(3) Let $\varphi$ be a cutoff function with support in $U=(0,1)\times\partial
M$, satisfying $\varphi\equiv 1$ on
$U_{\frac{1}{2}}:=(0,\frac{1}{2})\times\partial M\subset U$. For $u\in
C^{\infty}_{0}(M)$, one has $\varphi u\in C^{\infty}_{0}(U)$. Then Hardy
inequality (3.6) and Sobolev inequalities (3.3) imply

$\|r^{-l}|\nabla^{j}(\varphi
u)|\|_{C^{0}(U_{\frac{1}{2}})}\leq\|r^{-l}|\nabla^{j}(\varphi
u)|\|_{C^{0}(U)}\leq C\|r^{-l}|\nabla^{j}(\varphi u)|\|_{W^{k-l-j,p}(U)}\leq
C\|\varphi u\|_{W^{k,p}(U)}.$

Thus

$\|r^{-l}|\nabla^{j}u|\|_{C^{0}(U_{\frac{1}{2}})}\leq C\|u\|_{W^{k,p}(M)},$

since $\varphi\equiv 1$ on $U_{\frac{1}{2}}$. This completes the proof of (3)
in Proposition 3.1. ∎

###  3.2. Weighted Sobolev spaces and embedding

In this subsection, we review some weighted Sobolev spaces and derive a
weighted Sobolev inequality from Sobolev inequality in Proposition 3.1.

Let $(M^{n},g)$ be a compact Riemannian manifold with non-isolated conical
singularities as defined in Definition 2.1. For each $p\geq 1$,
$k\in\mathbb{N}$ and $\delta\in\mathbb{R}$, the weighted Sobolev space
$W^{k,p}_{\delta}(M)$ is the completion of $C^{\infty}_{0}(\mathring{M})$ with
respect to the weighted Sobolev norm

(3.11)
$\|u\|_{W^{k,p}_{\delta}(M)}=\left(\int_{M}\left(\sum^{k}_{i=0}\chi^{p(\delta-i)+n}|\nabla^{i}u|^{p}_{g}\right)d{\rm
vol}_{g}\right)^{\frac{1}{p}},$

where $\nabla^{i}u$ denotes the $i$-times covariant derivative of the function
$u$ with respect to the metric $g$, and $\chi\in C^{\infty}(\mathring{M})$ is
a positive weight function satisfying

(3.12) $\chi(x)=\begin{cases}1,&\text{if}\ \ x\in M\setminus U,\\\
\frac{1}{r},&\text{if}\ \ r={\rm dist}(x,\partial M)<\frac{1}{10},\end{cases}$

and $0<(\chi(x))^{-1}\leq 1$ for all $x\in\mathring{M}$. Recall that
$U=(0,1)\times\partial M\subset M$ is a collar neighborhood of the boundary
$\partial M$.

By the definition of the weighted Sobolev norm in $(\ref{eqn: WSN})$, we
clearly have

(3.13) $\delta^{\prime}>\delta\ \ \Rightarrow\ \
W^{k,p}_{\delta^{\prime}}(M)\subset W^{k,p}_{\delta}(M).$

Note also that the integral $\int^{1}_{0}r^{2(\mu-\delta)-1}\left(\ln
r\right)^{p}dr$ is finite if and only if $\mu>\delta$. Therefore, for a smooth
function $u$ on $\mathring{M}$ satisfying

(3.14) $u=O\left(r^{\mu}\left(\ln r\right)^{p}\right),\ \ \text{as}\ \ r\to
0,$

we have

(3.15) $u\in W^{k,2}_{\delta}(M)\Longleftrightarrow\mu>\delta.$

The weighted Sobolev norm $\|\cdot\|_{W^{k,p}_{\delta}}$ defined in
$(\ref{eqn: WSN})$ is essentially the same as the weighted Sobolev norm
$\|\cdot\|_{\mathcal{H}^{k,p}_{\gamma}}$ with $\delta=\gamma-\frac{n}{p}$ in
[CLW12].

Besides the Sobolev inequality in Proposition 3.1, as in Lemma 3.1 in [DW20],
another interesting consequence of the Hardy inequality (3.6) is the following
equivalence between some of the weighted Sobolev norms in $(\ref{eqn: WSN})$
with certain weight indices and the usual Sobolev norms in $(\ref{eqn: Sobolev
norm})$.

######  Lemma 3.3.

Let $(M^{n},g)$ be a compact Riemannian manifold with non-isolated conical
singularities. For each $p>1$ and $k\ \in\mathbb{N}$ satisfying $pi\neq n$ for
$i=1,2,\cdots,k$, there exists a constant $C=C(g,n,p,k)$ such that

$\|u\|_{W^{k,p}(M)}\leq\|u\|_{W^{k,p}_{k-\frac{n}{p}}(M)}\leq
C\|u\|_{W^{k,p}(M)}.$

Consequently, $W^{k,p}_{k-\frac{n}{p}}(M)=W^{k,p}(M)$ for any $p>1$ and
$k\in\mathbb{N}$ satisfying $pi\neq n$ for all $i=1,2,\cdots,k$.

By Lemma 3.3 and the Sobolev inequalities in Proposition 3.1, one immediately
obtains some weighted Sobolev inequalities with special weight indices.
However, we also need weighed Sobolev inequalities with general indices, which
can be established by using the Sobolev inequalities in Proposition 3.1 as
follows.

######  Proposition 3.4.

Let $(M^{n},g)$ be a compact Riemannian manifolds with non-isolated conical
singularities. For any $\delta\in\mathbb{R}$, $1\leq p<n$, $0\leq l\leq k$,
and $q$ with $\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}>0$ and
$\frac{np}{n-ip}\neq f,i=0,\cdots,k-l-1$, there exists a constant $C$, such
that for any $u\in C^{\infty}_{0}(\mathring{M})$,

$\|u\|_{W^{l,q}_{\delta}}\leq C\|u\|_{W^{k,p}_{\delta}}.$

Consequently, there is a continuous embedding

$W^{k,p}_{\delta}(M)\subset W^{l,q}_{\delta}(M).$

###### Proof.

For any $u\in C^{\infty}_{0}(\mathring{M})$,
$\chi^{\delta-i+\frac{n}{q}}|\nabla^{i}u|\in C^{\infty}_{0}(\mathring{M})$ for
any $i\in\mathbb{N}$ and $q>1$. Recall that $\chi$ is a weight function in
$(\ref{eqn: WeightFunction-cone})$. Then for $q_{1}=\frac{np}{n-p}$, i.e.
$\frac{1}{q_{1}}=\frac{1}{p}-\frac{1}{n}$, the Sobolev inequality in
Proposition 3.1 implies

$\displaystyle\|u\|_{W^{k-1,q_{1}}_{\delta}(M)}$ $\displaystyle=$
$\displaystyle\left(\int_{M}\left(\sum^{k-1}_{i=0}\chi^{q_{1}(\delta-i)+n}|\nabla^{i}u|^{q_{1}}_{g}\right)d{\rm
vol}_{g}\right)^{\frac{1}{q_{1}}}$ $\displaystyle\leq$ $\displaystyle
C\sum^{k-1}_{i=0}\|\chi^{\delta-i+\frac{n}{q_{1}}}|\nabla^{i}u|_{g}\|_{L^{q_{1}}}$
$\displaystyle\leq$ $\displaystyle
C\sum^{k-1}_{i=0}\|\chi^{\delta-i+\frac{n}{q_{1}}}|\nabla^{i}u|_{g}\|_{W^{1,p}(M)}$
$\displaystyle\leq$ $\displaystyle
C\sum^{k-1}_{i=0}\left(\|\nabla\left(\chi^{\delta-i+\frac{n}{q_{1}}}|\nabla^{i}u|_{g}\right)\|_{L^{p}(M)}+\|\chi^{\delta-i+\frac{n}{q_{1}}}|\nabla^{i}u|_{g}\|_{L^{p}(M)}\right)$
$\displaystyle\leq$ $\displaystyle
C\sum^{k-1}_{i=0}\big{(}\|\chi^{\delta-i+\frac{n}{q_{1}}+1}|\nabla^{i}u|_{g}\|_{L^{p}(M)}+\|\chi^{\delta-i+\frac{n}{q_{1}}}|\nabla^{i+1}u|_{g}\|_{L^{p}(M)}$
$\displaystyle+\|\chi^{\delta-i+\frac{n}{q_{1}}}|\nabla^{i}u|_{g}\|_{L^{p}(M)}\big{)}$
$\displaystyle\leq$ $\displaystyle
C\sum^{k}_{i=0}\|\chi^{\delta-i+\frac{n}{p}}|\nabla^{i}u|_{g}\|_{L^{p}(M)}\leq
C\|u\|_{W^{k,p}_{\delta}(M)}.$

In the third last inequality, we used $|\nabla\chi|_{g}\leq C\chi^{2}$. In the
second last inequality, we used $0<\chi^{-1}\leq 1$, and
$\frac{n}{q_{1}}=\frac{n}{p}-1$.

Then for any $0\leq l\leq k$, one can iterate the above inequality to complete
the proof. ∎

######  Remark 3.5.

The local version of the weighted Sobolev inequalities in Proposition 3.4 has
been estiblished in Proposition 3.2 and 3.3 in [CLW12] in a different manner.

######  Proposition 3.6.

Let $(M^{n},g)$ be a compact Riemannian manifold with non-isolated conical
singularities. For any $\delta\in\mathbb{R}$ and $k,l\in\mathbb{N}$ with
$k-l>\frac{n}{p}$, any $u\in W^{k,p}_{\delta}(M)$ satisfies

$|\nabla^{l}u|=o(r^{\delta-k+\frac{n}{p}}),$

as $r\rightarrow 0$.

###### Proof.

First note that if $u\in W^{k,p}_{\delta}(M)$ then
$\chi^{\delta-k+\frac{n}{p}}|\nabla^{l}u|\in W^{k-l,p}(M)$, where $\chi$ is
the weight function in (3.12) and it is equal to $\frac{1}{r}$ near the
singular set. Let
$U_{j}=\left(\left(\frac{1}{2}\right)^{j},\left(\frac{1}{2}\right)^{j-1}\right)\times\partial
M$ for each $j\in\mathbb{N}$. The Sobolev inequality in Proposition 3.1 gives

$\|r^{-\delta+k-\frac{n}{p}}|\nabla^{l}u|\|_{C^{0}(U_{j})}\leq
C\|r^{-\delta+k-\frac{n}{p}}|\nabla^{l}u|\|_{W^{k-l,p}(U_{j})}\leq
C\|u\|_{W^{k,p}_{\delta}(U_{j})}.$

Then because
$\sum\limits^{\infty}_{j=1}\|u\|_{W^{k,p}_{\delta}(U_{j})}\leq\|u\|_{W^{k,p}_{\delta}(M)}\leq\infty$,
$\|u\|_{W^{k,p}_{\delta}(U_{j})}\rightarrow 0$, as $j\rightarrow\infty$. Thus
$|\nabla^{l}u|=o(r^{\delta-k+\frac{n}{p}})$ as $r\rightarrow 0$. ∎

###  3.3. Compact weighted Sobolev embeddings

In this subsection, we prove that weighted Sobolev spaces
$W^{k,2}_{k-\frac{n}{2}}(M)$ can be compactly embedded into $L^{2}(M)$ space.
This compact embedding plays a crucial rule in the study of the spectrum of
the operator $-4\Delta_{g}+R_{g}$.

Firstly, we derive a compact embedding on the collar neighborhood:
$U=(0,1)\times\partial M$ of the singular set $\partial M$ with the model
metric $g_{0}$ given in $(\ref{eqn: model conical metric})$.

######  Lemma 3.7.

The continuous embedding

$i:W^{k,2}_{k-\frac{n}{2}}(U,g_{0})\hookrightarrow L^{2}(U,g_{0})$

is compact for each $k\in\mathbb{N}$.

The proof of Lemma 3.7 is similar in spirit as the proof of Lemma 3.2 in
[DW18] for isolated conical singularities. Roughly speaking, one relates the
problem to that on a cylinder. In [DW18], we did a spectral decomposition for
functions on a cone with respect to eigenfunctions of Laplacian on the cross
section, when deriving the inequality $(\ref{eqn: cone cylinder inequality})$
below. In the non-isolated conical singularities case, where we have a family
of cones, we cannot do the spectral decomposition as in [DW18], and we derive
$(\ref{eqn: cone cylinder inequality})$ without using the spectral
decomposition. So we provide details here.

###### Proof.

Clearly,
$\|u\|_{W^{k,2}_{k-\frac{n}{2}}(U,g_{0})}\geq\|u\|_{W^{l,2}_{l-\frac{n}{2}}(U,g_{0})}$,
for $k\geq l\in\mathbb{N}$. Therefore, it suffices to show that the embedding:
$i:W^{1,2}_{1-\frac{n}{2}}(C_{\epsilon}(N))\hookrightarrow
L^{2}(C_{\epsilon}(N))$, is compact.

Let $(C(\partial M),g_{Cyl})=((0,1)\times\partial M,dr^{2}+g_{\partial M})$ be
a finite cylinder. The usual Sobolev norm on the cylinder is given by

$\|u\|_{W^{1,2}(C(\partial M),g_{Cyl})}=\int_{C(\partial M)}(|\nabla
u|^{2}_{g_{Cyl}}+u^{2})dvol_{g_{Cyl}}.$

Note that the mapping

$\displaystyle L^{2}(U,g_{0})$ $\displaystyle\longrightarrow L^{2}(C(\partial
M),g_{Cyl})$ $\displaystyle u$
$\displaystyle\longmapsto\tilde{u}=r^{\frac{f}{2}}u,$

is unitary. We will show that

(3.16)
$\|u\|_{W^{1,2}_{1-\frac{n}{2}}(U,g_{0})}\geq\frac{3}{4}\|\tilde{u}\|_{W^{1,2}(C(\partial
M),g_{Cyl})},$

for all $u\in C^{\infty}_{0}(U)$. This then completes the proof, since the
embedding

$\mathring{W}^{1,2}(C(\partial M),g_{Cyl})\hookrightarrow L^{2}(C(\partial
M),g_{Cyl})$

is compact by the classical Rellich Lemma.

Now we prove the inequality $(\ref{eqn: cone cylinder inequality})$. For any
$u\in C^{\infty}_{0}(U)$,

$\displaystyle\|u\|^{2}_{W^{1,2}_{1-\frac{n}{2}}(U,g_{0})}$ $\displaystyle=$
$\displaystyle\int_{U}\left(\frac{1}{r^{2}}u^{2}+|\nabla
u|^{2}_{g_{0}}\right)d{\rm vol}_{g_{0}}$ $\displaystyle\geq$
$\displaystyle\int^{1}_{0}\int_{\partial
M}\left(\frac{1}{r^{2}}u^{2}+\left|\frac{\partial}{\partial
r}u\right|^{2}+|\nabla u|^{2}_{g_{\partial M}}\right)r^{f}d{\rm
vol}_{g_{\partial M}}dr$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\int_{\partial
M}\left(\frac{1}{r^{2}}\tilde{u}^{2}+\left|\frac{\partial}{\partial
r}\tilde{u}\right|^{2}-\frac{f^{2}}{4}\frac{1}{r^{2}}\tilde{u}^{2}+|\nabla\tilde{u}|^{2}_{g_{\partial
M}}\right)d{\rm vol}_{g_{\partial M}}dr$ $\displaystyle-\,f\int_{\partial
M}\int^{1}_{0}r^{f-1}u\left(\frac{\partial}{\partial r}u\right)drd{\rm
vol}_{g_{\partial M}}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\int_{\partial
M}\left(\frac{1}{r^{2}}\tilde{u}^{2}+\left|\frac{\partial}{\partial
r}\tilde{u}\right|^{2}-\frac{f^{2}}{4}\frac{1}{r^{2}}\tilde{u}^{2}+|\nabla\tilde{u}|^{2}_{g_{\partial
M}}\right)d{\rm vol}_{g_{\partial M}}dr$
$\displaystyle+\,\frac{f(f-1)}{2}\int^{1}_{1}\int_{\partial
M}\frac{1}{r^{2}}\tilde{u}^{2}d{\rm vol}_{g_{\partial M}}dr$ $\displaystyle=$
$\displaystyle\int_{C(\partial
M)}\left(\frac{1}{r^{2}}\left(1+\frac{f(f-2)}{4}\right)\tilde{u}^{2}+|\nabla\tilde{u}|^{2}_{g_{Cyl}}\right)d{\rm
vol}_{g_{Cyl}}$ $\displaystyle\geq$ $\displaystyle\int_{C(\partial
M)}\left(\frac{1}{r^{2}}\frac{3}{4}\tilde{u}^{2}+|\nabla\tilde{u}|^{2}_{g_{Cyl}}\right)d{\rm
vol}_{g_{Cyl}}.$ $\displaystyle\geq$
$\displaystyle\frac{3}{4}\|\tilde{u}\|^{2}_{W^{1,2}(C(\partial M),g_{Cyl})}.$

∎

Then by the asymptotic control for the metric $g$ on the collar neighborhood
$U$ in condition $(3)$ of Definition 2.1, on a sufficiently small collar
neighborhood $U_{\epsilon}=(0,\epsilon)\times\partial M$, metrics $g$ and
$g_{0}$ are quasi-isometric to each other, and so the corresponding weighted
Sobolev norms are equivalent. Thus, Lemma 3.7 implies the following compact
embedding property.

######  Proposition 3.8.

Let $(M^{n},g)$ be a compact manifold with non-isolated conical singularities
as in Definition 2.1. The continuous embedding

$i:W^{k,2}_{k-\frac{n}{2}}(M)\hookrightarrow L^{2}(M)$

is compact for each $k\in\mathbb{N}$.

We end this subsection by stating another compact weighted Sobolev embedding
that will be used in proving the existence of a minimizer of $W$-functional.

######  Proposition 3.9.

Let $(M^{n},g)$ be a compact Riemannian manifold with non-isolated conical
singularities. The embedding $W^{1,1}_{1-n}(M)\subset L^{q}(M)$ is compact for
any $1\leq q\leq\frac{n}{n-1}$.

This proposition can be derived similarly as in the proof of Proposition 3.5
in [DW20], except here we need to use the finite open cover as in the proof of
Lemma 3.2 for a collar neighborhood of $\partial M$. Thus we omit the proof.

###  3.4. Weighted elliptic estimate

In this subsection, we establish a weighted elliptic estimate on compact
manifolds with non-isolated conical singularities in Proposition 3.10. This
will be used in the derivation of asymptotic behavior of eigenfunctions of
Schrödinger operator $L:=-4\Delta+R$ and the minimizer of W-functional.

######  Proposition 3.10.

Let $(M,g)$ be a compact manifold with non-isolated conical singularities as
in Definition 2.1. If $u\in W^{k+1,2}_{\delta}(M)\cap C^{k+2}(\mathring{M})$
and $Lu\in W^{k,2}_{\delta-2}(M)$, then

(3.17) $\|u\|_{W^{k+2,2}_{\delta}(M)}\leq
C\left(\|Lu\|_{W^{k,2}_{\delta-2}(M)}+\|u\|_{W^{k+1,2}_{\delta}(M)}\right)$

holds for some constant $C=C(M,k,\delta)$ independent of function $u$.

######  Remark 3.11.

Note that this is slightly weaker than the usual elliptic estimate but it
suffices for our purpose.

###### Proof.

Clearly, it suffices to prove that there exists a constant $C=C(M,k,\delta)$
such that

(3.18) $\|u\|_{W^{k+2,2}_{\delta}(M)}\leq C_{k}\left(\|\Delta
u\|_{W^{k,2}_{\delta-2}(M)}+\|u\|_{W^{k+1,2}_{\delta}(M)}\right)$

holds for any $u\in W^{k+1,2}_{\delta}(M)\cap C^{k+2}(\mathring{M})$ with
$\Delta u\in W^{k,2}_{\delta-2}(M)$, as

(3.19) $\displaystyle\|\Delta u\|_{W^{k,2}_{\delta-2}(M,g)}$
$\displaystyle\leq\frac{1}{4}\|Lu\|_{W^{k,2}_{\delta-2}(M)}+\frac{1}{4}\|Ru\|_{W^{k,2}_{\delta-2}(M,g)}$
$\displaystyle\leq
C\left(\|Lu\|_{W^{k,2}_{\delta-2}(M,g)}+\|u\|_{W^{k,2}_{\delta}(M,g)}\right).$

To prove the inequality $(\ref{eqn: laplace elliptic estimate})$, we only need
to show it for the collar neighborhood $U=(0,1)\times\partial M$, that is

(3.20) $\|u\|_{W^{k+2,2}_{\delta}(U,g)}\leq C_{k}\left(\|\Delta
u\|_{W^{k,2}_{\delta-2}(U,g)}+\|u\|_{W^{k+1,2}_{\delta}(U,g)}\right),$

since the estimate on the smooth interior part $M\setminus U$ is classical.

In the rest of the proof, we prove the inequality $(\ref{eqn: cone laplace
elliptic estimate})$ by two steps. In the first step, we prove the inequality
$(\ref{eqn: cone laplace elliptic estimate})$ for $u\in C^{\infty}_{0}(U)$.
This is done by direct computation, using integration by parts as well as
commuting the derivatives. In the second step, we prove the inequality
$(\ref{eqn: cone laplace elliptic estimate})$ for functions $u\in
C^{k+2}(U)\cap W^{k+2,2}_{\delta}(U,g)$ with $\Delta u\in
W^{k,2}_{\delta-2}(U,g)$, by doing a cut-off and an approximation argument.

Step 1. Let $u\in C^{\infty}_{0}(U)$. We have:

$\displaystyle\|u\|^{2}_{W^{k+2,2}_{\delta}(U,g)}$ $\displaystyle=$
$\displaystyle\int_{U}\left(\frac{1}{r}\right)^{2(\delta-k-2)+n}\sum^{n}_{i_{1},\cdots,i_{k+2}=1}(\nabla_{i_{1}}\cdots\nabla_{i_{k+2}}u)(\nabla_{i_{1}}\cdots\nabla_{i_{k+2}}u)d{\rm
vol}_{g}$ $\displaystyle+\|u\|^{2}_{W^{k+1,2}_{\delta}(U,g)}.$

Here $\nabla_{i_{j}}$ denotes the covariant derivative $\nabla_{e_{i_{j}}}$
with respect to the metric $g$, where
$\left\\{e_{1},e_{2},\cdots,e_{n}\right\\}$ is a local orthornormal frame on
$(U,g)$. By doing the integration by parts for the integral in the first term
on the right hand side, one obtains

$\displaystyle\|u\|^{2}_{W^{k+2,2}_{\delta}(U,g)}$ $\displaystyle=$
$\displaystyle-\int_{U}\sum^{n}_{i_{1},\cdots,i_{k+2}=1}\nabla_{i_{1}}\left(\frac{1}{r}\right)^{2(\delta-k-2)+n}(\nabla_{i_{1}}\cdots\nabla_{i_{k+2}}u)(\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}u)d{\rm
vol}_{g}$
$\displaystyle-\int_{U}\left(\frac{1}{r}\right)^{2(\delta-k-2)+n}\sum^{n}_{i_{1},\cdots,i_{k+2}=1}(\nabla_{i_{1}}\nabla_{i_{1}}\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}u)(\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}u)d{\rm
vol}_{g}$
$\displaystyle+\int_{U}\left(\frac{1}{r}\right)^{2(\delta-k-2)+n}\sum^{n}_{i_{1},\cdots,i_{k+2}=1}(\nabla_{\nabla_{e_{i_{1}}}e_{i_{1}}}\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}u)(\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}u)d{\rm
vol}_{g}$ $\displaystyle+\|u\|^{2}_{W^{k+1,2}_{\delta}(U,g)}.$

Then one can switch $\nabla_{i_{1}}$ with
$\nabla_{i_{2}},\cdots,\nabla_{i_{k+2}}$ in the second term on the right hand
side to obtain $\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}(\Delta u)$, with some
additional terms involving the Riemann curvature tensor and its derivatives.
Note that the $i^{th}$ derivatives of curvature can be controlled by
$C_{i}\left(\frac{1}{r}\right)^{2+i}$ for some constant $C_{i}$. So by Cauchy-
Schwarz inequality, one has

(3.21)
$\displaystyle\|u\|^{2}_{W^{k+2,2}_{\delta}(U,g)}\leq\frac{1}{4}\|u\|^{2}_{W^{k+2,2}_{\delta}(U,g)}+C\|u\|^{2}_{W^{k+1,2}_{\delta}(U,g)}$
$\displaystyle\qquad-\int_{U}\left(\frac{1}{r}\right)^{2(\delta-k-2)+n}\sum^{n}_{i_{2},\cdots,i_{k+2}=1}(\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}(\Delta
u))(\nabla_{i_{2}}\cdots\nabla_{i_{k+2}}u)d{\rm vol}_{g}.$

Again, integrating by parts for the third term on the right hand side and
switching the order of covariant derivatives, one obtains similarly

(3.22) $\displaystyle\qquad\|u\|^{2}_{W^{k+2,2}_{\delta}(U,g)}$
$\displaystyle\leq\frac{1}{2}\|u\|^{2}_{W^{k+2,2}_{\delta}(U,g)}+C\|u\|^{2}_{W^{k+1,2}_{\delta}(U,g)}$
$\displaystyle\qquad+\int_{U}\left(\frac{1}{r}\right)^{2(\delta-k-2)+n}\sum^{n}_{i_{3},\cdots,i_{k+2}=1}(\nabla_{i_{3}}\cdots\nabla_{i_{k+2}}(\Delta
u))(\nabla_{i_{3}}\cdots\nabla_{i_{k+2}}(\Delta u))d{\rm vol}_{g}$
$\displaystyle\leq\frac{1}{2}\|u\|^{2}_{W^{k+2,2}_{\delta}(U,g)}+C\|u\|^{2}_{W^{k+1,2}_{\delta}(U,g)}+\|\Delta
u\|_{W^{k,2}_{\delta-2}(U,g)}.$

Finally, by rearranging the inequality, one obtains

(3.23) $\|u\|_{W^{k+2,2}_{\delta}(U,g)}\leq C\left(\|\Delta
u\|_{W^{k,2}_{\delta-2}(U,g)}+\|u\|_{W^{k+1,2}_{\delta}(U,g)}\right),$

for some constant $C$ and any $u\in C^{\infty}_{0}(U)$.

Step 2. Take a cut-off function $\varphi(r):(0,+\infty)\rightarrow[0,1]$ such
that

(3.24) $\varphi(r)=\begin{cases}0,&r\leq 1,\\\ 1,&r\geq 2,\end{cases}$

$0\leq\varphi\leq 1$ on $(1,2)$, and $|\varphi^{(k)}|\leq C_{k}$ for each
$k\in\mathbb{N}$ and some constant $C_{k}$.

For each $i\in\mathbb{N}\cup\\{0\\}$, let $\varphi_{i}(r):=\varphi(2^{i+1}r)$.
Then

(3.25) $\varphi_{i}(r)=\begin{cases}0,&r\leq\left(\frac{1}{2}\right)^{i+1},\\\
1,&r\geq\left(\frac{1}{2}\right)^{i},\end{cases}$

$0\leq\varphi_{i}\leq 1$, on $(2^{-(i+1)},2^{-i})$, and

$|\varphi^{(k)}_{i}(r)|\leq C_{k}\left(\frac{1}{r}\right)^{k},$

for any $i,k\in\mathbb{N}\cup\\{0\\}$ and a constant $C_{k}$.

For any $u\in C^{k+2}(U)\cap W^{k+1,p}_{\delta}(U,g)$ with $\Delta u\in
W^{k,2}_{\delta-2}(U,g)$, let $u_{i}=\varphi_{i}(r)u\in C^{k+2}_{0}(U)$. Then
one can check that

$u_{i}\rightarrow u\ \ \text{in}\ \ W^{k+1,2}_{\delta}(U,g),\quad\text{as}\ \
i\rightarrow\infty.$ $\Delta u_{i}\rightarrow\Delta u\ \ \text{in}\ \
W^{k,2}_{\delta-2}(U,g),\quad\text{as}\ \ i\rightarrow\infty.$

Consequently, we obtain that the inequality (3.20) holds for any $u\in
C^{k+2}(U)\cap W^{k+1,2}_{\delta}(U,g)$ with $\Delta u\in
W^{k,2}_{\delta-2}(U,g)$, since in step 1 we have shown that it holds for any
$u\in C^{\infty}_{0}(U,g)$. This completes the proof of the proposition. ∎

##  4\. Spectrum of $-4\Delta+R$ on compact manifolds with non-isolated
conical singularities

In this section, we study the spectrum of the Schrödinger operator
$-4\Delta+R$ on compact Riemannian manifolds with non-isolated conical
singularities. First we obtain a semi-boundedness estimate for the operator
$-4\Delta_{g_{0}}+R_{g_{0}}$ on a collar neighborhood of $\partial M$ in Lemma
4.1. This then implies a similar estimate for the operator
$-4\Delta_{g}+R_{g}$ in Proposition 4.2. Finally, we obtain the spectral
properties of the operator $-4\Delta+R$ on compact manifolds with non-isolated
conical singularities in Theorem 4.4.

In the following, we still set $L=-4\Delta_{g_{0}}+R_{g_{0}}$.

######  Lemma 4.1.

Let $(M^{n},g)$ be a compact manifold with non-isolated conical singularities
defined as in Definition 2.1 satisfying $\min\limits_{\partial
M}\\{R_{\hat{g}}\\}>(f-1)$. Then for a sufficiently small $\epsilon>0$, on the
collar neighborhood $U_{\epsilon}=(0,\epsilon)\times\partial M$ of the
singular set $\partial M$ with the model metric $g_{0}$, we have

$(Lu,u)_{L^{2}(U_{\epsilon},g_{0})}\geq\delta_{0}\|u\|^{2}_{W^{1,2}_{1-\frac{n}{2}}(U_{\epsilon},g_{0})},$

for all $u\in C^{\infty}_{0}(U_{\epsilon})$, and some $\delta_{0}>0$ depending
only on $\min\limits_{\partial M}\\{R_{\hat{g}}\\}$ and $f$.

###### Proof.

Because $\min\limits_{\partial M}\\{R_{\hat{g}}\\}>(f-1)$, and

(4.1) $f(f-1)+2\delta-\frac{4-\delta}{4}(f-1)^{2}\rightarrow f-1,\ \
\text{as}\ \ \delta\to 0,$

there exists a sufficiently small $\delta_{0}>0$ such that

(4.2) $\min\limits_{\partial
M}\\{R_{\hat{g}}\\}>f(f-1)+2\delta_{0}-\frac{4-\delta_{0}}{4}(f-1)^{2}.$

Then choose a sufficiently small $\epsilon$ such that on
$U_{\epsilon}=(0,\epsilon)\times\partial M$,

(4.3)
$\frac{\delta_{0}}{r^{2}}+R_{\check{g}}\circ\pi-|A|^{2}-|T|^{2}-|N|^{2}-2\check{\delta}N>0.$

Set

(4.4)
$L_{\delta_{0}}:=-(4-\delta_{0})\Delta_{g_{0}}+R_{g_{0}}-\frac{1}{r^{2}}\delta_{0}.$

Then we have

(4.5) $L=L_{\delta_{0}}-\delta_{0}\Delta_{g_{0}}+\frac{1}{r^{2}}\delta_{0},$

and for any $u\in C^{\infty}_{0}(U_{\epsilon})$,

(4.6) $\displaystyle(Lu,u)_{L^{2}(U_{\epsilon},g_{0})}$ $\displaystyle=$
$\displaystyle\int_{U_{\epsilon}}(L_{\delta_{0}}u)ud{\rm
vol}_{g_{0}}+\int_{U_{\epsilon}}\left((-\delta_{0}\Delta_{g_{0}}u)u+\frac{1}{r^{2}}\delta_{0}u^{2}\right)d{\rm
vol}_{g_{0}}$ $\displaystyle=$
$\displaystyle\int_{U_{\epsilon}}(L_{\delta_{0}}u)ud{\rm
vol}_{g_{0}}+\delta_{0}\int_{U_{\epsilon}}\left(|\nabla
u|^{2}_{g_{0}}+\frac{1}{r^{2}}u^{2}\right)d{\rm vol}_{g_{0}}$ $\displaystyle=$
$\displaystyle(L_{\delta_{0}}u,u)_{L^{2}(U_{\epsilon},g_{0})}+\delta_{0}\|u\|^{2}_{W^{1,2}_{1-\frac{n}{2}}(U_{\epsilon},g_{0})}.$

Thus, it suffices to show that
$(L_{\delta_{0}}u,u)_{L^{2}(U_{\epsilon},g_{0})}\geq 0$ to complete the proof.
Indeed, we claim that:

(4.7) $(L_{\delta_{0}}u,u)_{L^{2}(U_{\epsilon},g_{0})}\geq
C\|u\|^{2}_{L^{2}(U_{\epsilon},g_{0})},$

holds for any $u\in C^{\infty}_{0}(U_{\epsilon})$, where

(4.8) $C=\min\left\\{(4-\delta_{0})\frac{(f-1)^{2}}{4}+\min\limits_{\partial
M}\\{R_{\hat{g}}\\}-2\delta_{0}-f(f-1),1\right\\}>0.$

In the rest of the proof, we prove the claim in $(\ref{eqn: L estimate
claim})$. We start with:

(4.9) $\displaystyle(L_{\delta_{0}}u,u)_{L^{2}(U_{\epsilon},g_{0})}$
$\displaystyle=$ $\displaystyle\int_{U_{\epsilon}}\left((4-\delta_{0})|\nabla
u|^{2}_{g_{0}}+\left(R_{g_{0}}-\frac{\delta_{0}}{r^{2}}\right)u^{2}\right)d{\rm
vol}_{g_{0}}$ $\displaystyle\geq$
$\displaystyle\int_{U_{\epsilon}}\left((4-\delta_{0})\left|\frac{\partial}{\partial
r}u\right|^{2}+\left(R_{g_{0}}-\frac{\delta_{0}}{r^{2}}\right)u^{2}\right)r^{f}d{\rm
vol}_{g_{\partial M}}dr$

Setting $\tilde{u}:=r^{\frac{f}{2}}u$, then the right hand side of $(\ref{eqn:
L_delta estimate})$ can be rewritten as

$\displaystyle\int^{\epsilon}_{0}\int_{\partial
M}\left((4-\delta_{0})\left|\frac{\partial}{\partial
r}\tilde{u}\right|^{2}-\frac{(4-\delta_{0})f^{2}}{4r^{2}}\tilde{u}^{2}+\left(R_{g_{0}}-\frac{\delta_{0}}{r^{2}}\right)\tilde{u}^{2}\right)drd{\rm
vol}_{g_{\partial M}}$ $\displaystyle-\,(4-\delta_{0})f\int_{\partial
M}\int^{\epsilon}_{0}r^{f-1}\tilde{u}\left(\frac{\partial}{\partial
r}\tilde{u}\right)drd{\rm vol}_{g_{\partial M}}$

It simplifies via integration by parts to

$\displaystyle\int^{\epsilon}_{0}\int_{\partial
M}\Bigg{[}(4-\delta_{0})\left|\frac{\partial}{\partial
r}\tilde{u}\right|^{2}+(4-\delta_{0})\frac{f(f-2)}{4}\frac{1}{r^{2}}\tilde{u}^{2}$
(4.10)
$\displaystyle+\left(R_{g_{0}}-\frac{\delta_{0}}{r^{2}}\right)\tilde{u}^{2}\Bigg{]}d{\rm
vol}_{g_{\partial M}}dr.$

Now apply the one dimensional Hardy inequality $(\ref{eqn: Hardy inequality})$
to the first term and use the formula (2.5) for the scalar curvature
$R_{g_{0}}$ combined with (4.3). Then the quantity in (4.10) is

(4.11) $\displaystyle\geq$ $\displaystyle\int^{\epsilon}_{0}\int_{\partial
M}\Big{[}\frac{1}{4}(4-\delta_{0})\frac{1}{r^{2}}\tilde{u}^{2}+(4-\delta_{0})\frac{f(f-2)}{4}\frac{1}{r^{2}}\tilde{u}^{2}$
$\displaystyle+\,\frac{1}{r^{2}}(R_{\hat{g}}-2\delta_{0}-f(f-1))\tilde{u}^{2}\Big{]}d{\rm
vol}_{g_{\partial M}}dr$ $\displaystyle+\,\int^{\epsilon}_{0}\int_{\partial
M}\left(\frac{\delta_{0}}{r^{2}}+R_{\check{g}}\circ\pi-|A|^{2}-|T|^{2}-|N|^{2}-\check{\delta}N\right)d{\rm
vol}_{g_{\partial M}}dr$ $\displaystyle\geq$
$\displaystyle\int^{\epsilon}_{0}\int_{\partial
M}\left((4-\delta_{0})\frac{(f-1)^{2}}{4}+\min\limits_{\partial
M}\\{R_{\hat{g}}\\}-2\delta_{0}-f(f-1)\right)\frac{1}{r^{2}}\tilde{u}^{2}d{\rm
vol}_{g_{\partial M}}dr$ $\displaystyle\geq$ $\displaystyle
C\int^{\epsilon}_{0}\int_{\partial M}\tilde{u}^{2}d{\rm vol}_{g_{\partial
M}}dr=C\|u\|^{2}_{L^{2}(U_{\epsilon},g_{0})}.$

Here the constant $C$ is given by $(\ref{eqn: L estimate C})$. Combining
(4.9), (4.10), (4.11), this proves the claim in $(\ref{eqn: L estimate
claim})$, and completes the proof of the lemma. ∎

Then because on a sufficiently small collar neighborhood of the boundary the
metric $g$ is uniformly equivalent to $g_{0}$ and the manifold is compact,
Lemma 4.1 implies the following semi-boundedness estimate on the manifold with
non-isolated conical singularities.

######  Proposition 4.2.

Let $(M^{n},g)$ be a compact Riemannian manifold with non-isolated conical
singularities with $\inf\limits_{\partial M}\\{R_{\hat{g}}\\}>(f-1)$. Then
there exists a large enough constant $A$ such that the operator
$L_{A}:=-4\Delta_{g}+R_{g}+A$ satisfies

(4.12) $(L_{A}u,u)_{L^{2}(M,g)}\geq
C\|u\|^{2}_{W^{1,2}_{1-\frac{n}{2}}(M,g)},$

for any $u\in C^{\infty}_{0}(\mathring{M})$ and some constant $C>0$. In
particular, the operator $L_{A}$ with the domain ${\rm
Dom}(L_{A})=C^{\infty}_{0}(\mathring{M})$ is strictly positive.

######  Remark 4.3.

The constant $A$ in Proposition 4.2 is need to control the scalar curvature of
the metric $g$ outside of a collar neighborhood of the boundary, since we do
not impose any restrictions for it.

######  Theorem 4.4.

Let $(M,g)$ be a compact Riemannian manifold with non-isolated conical
singularities with $\inf\limits_{\partial M}\\{R_{\hat{g}}\\}>(f-1)$. Then the
spectrum of the Friedrichs extension of the operator $-4\Delta_{g}+R_{g}$ with
domain $C^{\infty}_{0}(\mathring{M})$ consists of discrete eigenvalues with
finite multiplicities

(4.13) $\lambda_{1}\leq\lambda_{2}\leq\lambda_{2}\leq\cdots,$

and $\lambda_{k}\rightarrow\infty$. The corresponding eigenfunctions
$\\{u_{i}\\}^{\infty}_{i=1}$ form a complete basis of $L^{2}(M)$. Moreover,
eigenfunctions $u_{i}$ are smooth on $\mathring{M}$ and satisfy the usual
eigenfunction equation on $\mathring{M}$.

###### Proof.

The existence of the self-adjoint, strictly positive and surjective Friedrichs
extension $\tilde{L}_{A}$ with domain ${\rm Dom}(\tilde{L}_{A})$ of the
operator $L_{A}$ in Proposition 4.2 follows from the Neumann Theorem in [EK96]
and the estimate in Proposition 4.2. Furthermore, Proposition 4.2 also implies
that the completion of $C^{\infty}_{0}(\mathring{M})$ with respect to the norm
$\|u\|_{L_{A}}:=(L_{A}u,u)_{L^{2}(M,g)}$ is contained in the weighted Sobolev
space $W^{1,2}_{1-\frac{n}{2}}(M)$. Thus from the construction of the
Friedrichs extension in the proof of the Neumann theorem in [EK96], one can
see that ${\rm Dom}(\tilde{L}_{A})\subset H^{1}(M)$.

Because $\tilde{L}_{A}:{\rm Dom}(\tilde{L}_{A})\rightarrow L^{2}(M)$ is one-
to-one and onto, the inverse

(4.14) $\tilde{L}_{A}^{-1}:L^{2}(M)\rightarrow{\rm
Dom}(\tilde{L}_{A})\hookrightarrow W^{1,2}_{1-\frac{n}{2}}(M)\hookrightarrow
L^{2}(M)$

exists. This inverse map is self-adjoint and compact, since the embedding
$W^{1,2}_{1-\frac{n}{2}}(M)\hookrightarrow L^{2}(M)$ is compact by Proposition
3.8. Then the spectrum theorem of self-adjoint compact operators implies the
desired spectral properties of the Friedrichs extension of $L_{A}$, and hence
those for the Friedrichs extension of $L=-4\Delta_{g}+R_{g}$ stated in the
theorem.

Finally, the regularity of the eigenfunctions follows from the standard
elliptic regularity theory for weak solutions of eigenvalue equations, since
this is a local property. ∎

##  5\. Asymptotic behavior of eigenfunctions

In this section, we study the asymptotic behavior of eigenfunctions of
$-4\Delta+R$ on manifolds with non-isolated conical singularities and prove
Theorem 1.4 by a Nash-Moser iteration argument.

######  Proposition 5.1.

Let $(M^{n},g)$ be a $n$-dimensional compact manifold with non-isolated
conical singularities as in Definition 2.1, and $u$ an eigenfunction of
$-4\Delta_{g}+R_{g}$ with eigenvalue $\lambda$. Then $u$ satisfies

(5.1)
$\|u\|_{L^{\infty}\left(\left(\frac{1}{4}\epsilon,\frac{1}{2}\epsilon\right)\times\partial
M,g\right)}\leq
C\cdot\epsilon^{-\frac{n}{2}+1}\cdot\|u\|_{W^{2,2}_{1-\frac{n}{2}}\left(\left(\frac{1}{8}\epsilon,\epsilon\right)\times\partial
M,g\right)},$

and

(5.2) $\|\nabla
u\|_{L^{\infty}\left(\left(\frac{1}{4}\epsilon,\frac{1}{2}\epsilon\right)\times\partial
M,g\right)}\leq
C\cdot\epsilon^{-\frac{n}{2}}\cdot\|u\|_{W^{2,2}_{1-\frac{n}{2}}\left(\left(\frac{1}{8}\epsilon,\epsilon\right)\times\partial
M,g\right)},$

for some constant $C$ and any $\epsilon<1$.

###### Proof.

Let $w=|\nabla u|^{2}+\frac{1}{r^{2}}u^{2}$. We apply Nash-Moser iteration to
$w$, to obtain the estimates for $u$ in $(\ref{eqn: eigenfunction estimate})$
and $(\ref{eqn: eigenfunction gradient estimate})$.

Step 1. We derive the following differential inequality for $w$:

(5.3) $\Delta w\geq-|R|w-|\lambda|w-Cr^{-2}w.$

The Bochner formula implies

(5.4) $\begin{split}\frac{1}{2}\Delta|\nabla u|^{2}&=|{\rm
Hess}\,u|^{2}+\langle\nabla u,\nabla\Delta u\rangle+{\rm Ric}(\nabla u,\nabla
u)\\\ &\geq\frac{1}{4}\langle\nabla u,\nabla(Ru-\lambda u)\rangle+{\rm
Ric}(\nabla u,\nabla u)\\\ &=\frac{1}{4}R|\nabla
u|^{2}+\frac{1}{4}\langle\nabla u,\nabla R\rangle u-\frac{\lambda}{4}|\nabla
u|^{2}+{\rm Ric}(\nabla u,\nabla u)\\\
&\geq-\frac{1}{4}|R|w-\frac{|\lambda|}{4}w-C_{0}r^{-2}w-\frac{1}{4}|\nabla
u|\cdot|\nabla R|\cdot|u|,\end{split}$

since Ricci curvature is bounded by $C_{0}r^{-2}$. Note that $|\nabla R|\leq
C_{1}r^{-3}$ near the conically singular points for some positive constant
$C_{1}$. Thus,

(5.5) $\begin{split}\frac{1}{2}\Delta|\nabla
u|^{2}&\geq-\frac{1}{4}|R|w-\frac{|\lambda|}{4}w-C_{0}r^{-2}w-C_{1}\frac{1}{4}r^{-3}|\nabla
u|\cdot|u|\\\
&\geq-\frac{1}{4}|R|w-\frac{|\lambda|}{4}w-C_{0}r^{-2}w-C_{1}\frac{1}{8}r^{-2}(|\nabla
u|^{2}+r^{-2}u^{2})\\\
&=-\frac{1}{4}|R|w-\frac{|\lambda|}{4}w-C_{0}r^{-2}w-C_{1}\frac{1}{8}r^{-2}w\end{split}$

Moreover,

(5.6) $\begin{split}\Delta(r^{-2}u^{2})&=\Delta(r^{-2})u^{2}+2\langle\nabla
r^{-2},\nabla u^{2}\rangle+r^{-2}\Delta u^{2}\\\ &\geq
4r^{-4}u^{2}-4r^{-3}|\nabla
u|\cdot|u|-\frac{1}{2}|R|w-\frac{1}{2}|\lambda|w\\\ &\geq
2r^{-2}w-\frac{1}{2}|R|w-\frac{1}{2}|\lambda|w\end{split}$

By combining inequalities in $(\ref{eqn: Laplace estimate 1})$ and $(\ref{eqn:
Laplace estimate 2})$, we obtain the differential inequality in $(\ref{eqn:
Laplace estimate})$

Step 2. Iteration process.

For any $q>1$ and $\eta\in C^{\infty}_{0}(\mathring{M})$, straightforward
calculations give

(5.7) $\displaystyle\int|\nabla(\eta w^{\frac{q}{2}})|^{2}$ $\displaystyle=$
$\displaystyle\int w^{q}|\nabla\eta|^{2}-\frac{q}{2}\int\eta^{2}w^{q-1}\Delta
w$ $\displaystyle-\left(1-\frac{2}{q}\right)\int|\nabla(\eta
w^{\frac{q}{2}})|^{2}-\left(1-\frac{2}{q}\right)\int\eta w^{q}\Delta\eta.$

By rearranging the equation and then using the inequality (5.3), we obtain

$\displaystyle\int|\nabla(\eta w^{\frac{q}{2}})|^{2}$ $\displaystyle=$
$\displaystyle\frac{q}{2(q-1)}\int
w^{q}|\nabla\eta|^{2}-\frac{q^{2}}{4(q-1)}\int\eta^{2}w^{q-1}\Delta
w-\frac{q-2}{2(q-1)}\int\eta w^{q}\Delta\eta$ $\displaystyle\leq$
$\displaystyle\frac{q}{2(q-1)}\int
w^{q}|\nabla\eta|^{2}+\frac{q^{2}|\lambda|}{8(q-1)}\int\eta^{2}w^{q}+\frac{q^{2}}{8(q-1)}\int\eta^{2}w^{q}|R|$
$\displaystyle+\frac{q^{2}C}{2(q-1)}\int\eta^{2}r^{-2}w^{q}-\frac{q-2}{2(q-1)}\int\eta
w^{q}\Delta\eta$

Define $q_{k}:=\mu^{k}$, for $k\geq 0$, where $\mu=\frac{n}{n-2}$, and

(5.9) $\displaystyle a_{k}$ $\displaystyle:=$
$\displaystyle\left(\frac{1}{16}+\frac{1}{16}\sum^{k}_{i=0}\left(\frac{1}{2}\right)^{i}\right)\epsilon,$
(5.10) $\displaystyle b_{k}$ $\displaystyle:=$
$\displaystyle\left(\frac{5}{4}-\frac{1}{4}\sum^{k}_{i=0}\left(\frac{1}{2}\right)^{i}\right)\epsilon.$

Then let $U_{k}:=(a_{k},b_{k})\times\partial M\subset U$ with the restricted
metric $g|_{U_{k}}$. Choose cut-off functions $\eta_{k}:=\varphi_{k}(r)\in
C^{\infty}_{0}(U_{k})$ such that

(5.11) $\eta_{k}\equiv 1,\ \ \text{on}\ \
U_{k+1};\quad|\varphi^{\prime}_{k}|\leq
2^{k+5}\epsilon^{-1},\quad|\varphi^{\prime\prime}_{k}|\leq
2^{2k+10}\epsilon^{-2}.$

Now for any $k\geq 1$, applying the Sobolev inequality (with $l=0,k=1$, $p=2$
and $q=\frac{2n}{n-2}$) in Proposition 3.1 and substituting $\eta_{k}$ into
the inequality in (5) give

(5.12) $\displaystyle\left(\int_{U_{k+1}}w^{q_{k+1}}\right)^{\frac{1}{\mu}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{U_{k}}\left(\eta^{2}_{k}w^{q_{k}}\right)^{\mu}\right)^{\frac{1}{\mu}}$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{U_{k}}\left|\nabla\left(\eta_{k}w^{\frac{q_{k}}{2}}\right)\right|^{2}+\int_{U_{k}}\eta^{2}_{k}w^{q_{k}}\right)$
$\displaystyle\leq$ $\displaystyle
C\Bigg{(}\frac{q_{k}}{2(q_{k}-1)}\int_{U_{k}}w^{q_{k}}|\nabla\eta_{k}|^{2}+\frac{q_{k}^{2}|\lambda|}{8(q_{k}-1)}\int_{U_{k}}\eta^{2}_{k}w^{q_{k}}$
$\displaystyle+\frac{q_{k}^{2}}{8(q_{k}-1)}\int_{U_{k}}\eta^{2}_{k}w^{q_{k}}|R|+\frac{q_{k}^{2}C}{2(q_{k}-1)}\int_{U_{k}}\eta^{2}_{k}r^{-2}w^{q_{k}}$
$\displaystyle-\frac{q_{k}-2}{2(q_{k}-1)}\int_{U_{k}}\eta_{k}w^{q_{k}}\Delta\eta_{k}+\int_{U_{k}}r^{-2}\eta^{2}_{k}w^{q_{k}}\Bigg{)}$
$\displaystyle\leq$ $\displaystyle C\cdot(4+|\lambda|)\cdot q^{2}_{k}\cdot
2^{2k+10}\cdot\epsilon^{-2}\int_{U_{k}}w^{q_{k}}.$

Then by iterating the above process, one has

(5.13)
$\begin{split}\|w\|_{L^{q_{k+1}}(U_{k+1})}&\leq(C\cdot(4+|\lambda|))^{\frac{1}{q_{k}}}\cdot
q^{\frac{2}{q_{k}}}_{k}\cdot
2^{\frac{2k+10}{q_{k}}}\cdot\epsilon^{-2\frac{1}{q_{k}}}\cdot\|w\|_{L^{q_{k}}(U_{k})}\\\
&\leq\cdots\\\
&\leq(C\cdot(4+|\lambda|))^{\sum\limits^{k}_{i=1}\frac{1}{q_{i}}}\cdot\left(\prod^{k}_{i=1}q^{\frac{2}{q_{i}}}_{i}\right)\cdot
2^{\sum\limits^{k}_{i=1}\frac{2i+10}{q_{i}}}\cdot\epsilon^{-2\sum\limits^{k}_{i=1}\frac{1}{q_{i}}}\cdot\|w\|_{L^{q_{1}}(U_{1})}.\end{split}$

By letting $k\rightarrow\infty$ in the inequality in $(\ref{eqn: iteration
inequality})$, we obtain

(5.14) $\|w\|_{L^{\infty}(U_{\infty})}\leq
C\cdot\epsilon^{-(n-2)}\cdot\|w\|_{L^{q_{1}}(U_{1})}.$

Step 3. We estimate $\|w\|_{L^{q_{1}}(U_{1})}$ and complete the proof. By
using Sobolev inequality, we have

(5.15) $\displaystyle\||\nabla u|^{2}\|_{L^{q_{1}}(U_{1})}$ $\displaystyle=$
$\displaystyle\|\nabla u\|^{2}_{L^{\frac{2n}{n-2}}(U_{1})}$
$\displaystyle\leq$ $\displaystyle\|\eta_{0}|\nabla
u|\|^{2}_{L^{\frac{2n}{n-2}}(U_{0})}$ $\displaystyle\leq$ $\displaystyle
C\left(\|\nabla(\eta_{0}|\nabla u|)\|^{2}_{L^{2}(U_{0})}+\|\eta_{0}|\nabla
u|\|^{2}_{L^{2}(U_{0})}\right)$ $\displaystyle\leq$ $\displaystyle
C\left(\|\nabla^{2}u\|^{2}_{L^{2}(U_{0})}+\epsilon^{-2}\|\nabla
u\|^{2}_{L^{2}(U_{0})}\right).$

Because $u\in W^{1,2}_{1-\frac{n}{2}}(M)\cap C^{\infty}(\mathring{M})\subset
W^{0,2}_{1-\frac{n}{2}-2}$ and satisfies the eigenvalue equation $Lu=\lambda
u$, Proposition 3.10 implies that $u\in W^{2,2}_{1-\frac{n}{2}}(M)$. So

(5.16) $\displaystyle+\infty$ $\displaystyle>$
$\displaystyle\|u\|_{W^{2,2}_{1-\frac{n}{2}}(M)}$ $\displaystyle\geq$
$\displaystyle\|u\|_{W^{2,2}_{1-\frac{n}{2}}(U_{0})}$ $\displaystyle\geq$
$\displaystyle\left(\int_{U_{0}}\left(\frac{1}{r}\right)^{2(1-\frac{n}{2}-2)+n}|\nabla^{2}u|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\geq$ $\displaystyle C\epsilon\|\nabla^{2}u\|_{L^{2}(U_{0})}.$

This gives $\|\nabla^{2}u\|^{2}_{L^{2}(U_{0})}\leq
C\epsilon^{-2}\|u\|^{2}_{W^{2,2}_{1-\frac{n}{2}}(U_{0})}$. Then by substitute
this inequality into $(\ref{eqn: u gradient integral estimate})$, we obtain

(5.17) $\||\nabla u|^{2}\|^{2}_{L^{q_{1}}(U_{1})}\leq
C\epsilon^{-2}\|u\|^{2}_{W^{2,2}_{1-\frac{n}{2}}(U_{0})}.$

Moreover, using Sobolev inequality again gives

(5.18)
$\begin{split}\|r^{-2}u^{2}\|_{L^{q_{1}}(U_{1})}&=\|r^{-1}u\|^{2}_{L^{\frac{2n}{n-2}}(U_{1})}\\\
&\leq C\epsilon^{-2}\|\eta_{0}u\|^{2}_{L^{\frac{2n}{n-2}}(U_{0})}\\\ &\leq
C\epsilon^{-2}\|u\|^{2}_{W^{1,2}_{1-\frac{n}{2}}(U_{0})}.\end{split}$

By combining the inequalities in $(\ref{eqn: w estimate 1})$ and $(\ref{eqn: w
estimate 2})$, we obtain

(5.19) $\|w\|_{L^{q_{1}}(U_{1})}\leq
C\epsilon^{-2}\|u\|^{2}_{W^{2,2}_{1-\frac{n}{2}}(U_{0})}.$

Finally, by substituting the estimate in $(\ref{eqn: w integral estimate})$
into the inequality in $(\ref{eqn: iteration limit inequality})$, we finish
the proof. ∎

Now by setting $\epsilon=\left(\frac{1}{2}\right)^{j}$ for $j\geq 0$ in
Proposition 5.1, we have

(5.20)
$\left(\frac{1}{2}\right)^{\left(\frac{n}{2}-1\right)j}\|u\|_{L^{\infty}((2^{-(j+2)},\,2^{-(j+1)})\times\partial
M)}\leq
C\cdot\|u\|_{W^{2,2}_{1-\frac{n}{2}}((2^{-(j+3)},\,2^{-j})\times\partial
M)}\rightarrow 0,$

as $j\rightarrow\infty$, since
$\sum\limits^{\infty}_{j=1}\|u\|_{W^{2,2}_{1-\frac{n}{2}}((2^{-(j+3)},\,2^{-j})\times\partial
M)}\leq 3\|u\|_{W^{2,2}_{1-\frac{n}{2}}(M)}<\infty$.

The same arguement gives an estimate for $|\nabla u|$. We obtain

######  Corollary 5.2.

Let $(M^{n},g)$ be a $n$-dimensional compact manifold with non-isolated
conical singularities as in Definition 2.1, and $u$ an eigenfunction of
$-4\Delta_{g}+R_{g}$. Then $u$ satisfies

(5.21) $|\nabla^{i}u|=o(r^{-\frac{n}{2}+1-i})$

as $r\rightarrow 0$, for $i=0$ and $1$.

##  6\. Lower boundedness of the infimum of $W$-functional and asymptotic
behavior of the minimizer

By using the estimate in Proposition 4.2, Sobolev inequality in Proposition
3.1, and compactness of weighted Sobolev embeddings in Proposition 3.8 and
Proposition 3.9, one can obtain that the infimum of $W$-functional is finite
and a smooth positive function realizes the infimum, provided
$R_{h_{0}}>(f-1)$, via direct methods in the calculus of variations. The proof
is verbatim as in the proof of Proposition 4.2 in [DW18]. So we omit it here.

Moreover, the asymptotic estimate for the minimizer of the $W$-functional can
be obtained by combining weighted elliptic bootstrapping and Nash-Moser
iteration as we do for eigenfunctions of $-4\Delta+R$ in §5. However, unlike
the eigenfunction equation, the Euler-Lagrange equation of minimizer of
$W$-functional in (6.1) below is nonlinear. In the rest of the section, we
explain how to deal with new difficulties caused by the nonlinear term $u\ln
u$ in the derivation of asymptotic estimate for the minimizer $u$ in Theorem
1.6.

Recall that the Euler-Lagrange equation of the minimizer $u$ of $W$-functional
is given by, see e.g. the equation (4.17) in [DW20],

(6.1) $-4\Delta_{g}u+R_{g}u-\frac{2}{\tau}u\ln u-\frac{n+m}{\tau}u=0,$

with the constraint $\|u\|_{L^{2}(M)}=(4\pi\tau)^{\frac{n}{4}}$, where $m$ is
the infimum of the $W$-functional.

Let $v=|\nabla u|^{2}+r^{-2}u^{2}$. As in step 1 in the proof of Proposition
5.1, one easily sees that

(6.2) $\Delta(r^{-2}u^{2})\geq
2r^{-2}v-\frac{1}{2}|R|v-\frac{|n+m|}{8\tau}v-\frac{1}{\tau}r^{-2}u^{2}\ln u,$

and

(6.3) $\Delta|\nabla u|^{2}\geq-\frac{1}{2}|R|v-\frac{|n+m|}{\tau}v-Cv-
Cr^{-2}v-\frac{1}{\tau}|\nabla u|^{2}\ln u.$

So

(6.4) $\Delta v\geq-|R|v-Cv-Cr^{-2}v-\frac{1}{\tau}v\ln u.$

Then substituting $v$ into the inequality in (5) gives

$\displaystyle\int|\nabla(\eta v^{\frac{q}{2}})|^{2}$ $\displaystyle=$
$\displaystyle\frac{q}{2(q-1)}\int
v^{q}|\nabla\eta|^{2}-\frac{q^{2}}{4(q-1)}\int\eta^{2}v^{q-1}\Delta
v-\frac{q-2}{2(q-1)}\int\eta v^{q}\Delta\eta$ $\displaystyle\leq$
$\displaystyle\frac{q}{2(q-1)}\int
v^{q}|\nabla\eta|^{2}-\frac{q^{2}}{4(q-1)}\int\eta^{2}v^{q}|R|$
$\displaystyle+C\frac{q^{2}}{4(q-1)}\int\eta^{2}v^{q}+C\frac{q^{2}}{4(q-1)}\int\eta^{2}r^{-2}v^{q}$
$\displaystyle+\frac{q^{2}}{8(q-1)\tau}\int\eta^{2}v^{2}\ln
u-\frac{q-2}{2(q-1)}\int\eta^{2}v^{q}\Delta\eta,$

for any nonnegative $\eta\in C^{\infty}_{0}(\mathring{M})$ and $q>1$. Then the
only difference from the eigenfunction estimate in Proposition 5.1 is the term
involving the integral of $v^{q}\ln u$. So we briefly describe how to deal
with this term.

For a sufficiently small $\gamma<1$ (with $\frac{2}{\gamma}>\frac{n}{2}$),
there exists a constant $a$ such that

(6.6) $\displaystyle\int\eta^{2}v^{q}\ln u$ $\displaystyle=$
$\displaystyle\int_{\\{0<u\leq e\\}}\eta^{2}v^{q}\ln
u+\int_{\\{u>e\\}}\eta^{2}v^{q}\ln u$ (6.7) $\displaystyle\leq$
$\displaystyle\int_{\\{0<u\leq
e\\}}\eta^{2}v^{q}+a\int_{\\{u>e\\}}\eta^{2}v^{q}u^{\gamma}$ (6.8)
$\displaystyle\leq$
$\displaystyle\int\eta^{2}v^{q}+a\int\eta^{2}v^{q}u^{\gamma}.$

Then by Hölder inequality and Young’s inequality (e.g. as on p. 20 in
[DWZ18]),

(6.9) $\displaystyle a\frac{q^{2}}{8(q-1)\tau}\int\eta^{2}v^{q}u^{\gamma}$
$\displaystyle\leq$ $\displaystyle a\frac{q^{2}}{8(q-1)\tau}\left(\int
u^{2}\right)^{\frac{\gamma}{2}}\cdot\left(\int\left(\eta^{2}v^{q}\right)^{\frac{2}{2-\gamma}}\right)^{\frac{2-\gamma}{2}}$
(6.10) $\displaystyle\leq$ $\displaystyle
a(4\pi\tau)^{\frac{n\gamma}{4}}\left[\delta\left(\int(\eta^{2}v^{q})^{\mu}\right)^{\frac{1}{\mu}}+\delta^{-\frac{n\gamma}{4-n\gamma}}\cdot\left(\frac{q^{2}}{8(q-1)\tau}\right)^{\frac{4-2n\gamma}{4-n\gamma}}\left(\int\eta^{2}v^{q}\right)\right],$

for any small $\delta>0$. We may take $\delta$ so that $C\cdot
a\cdot(4\pi\tau)^{\frac{n\gamma}{4}}\cdot\delta=\frac{1}{2}$, where $C$ is the
$L^{2}$-Sobolev constant in Proposition 3.1.

Another remark is about the derivation of $u\in W^{0,2}_{1-\frac{n}{2}}(M)$ by
using Proposition 3.10. For this one needs to show that $u\ln u\in
W^{0,2}_{1-\frac{n}{2}-2}$, since $Lu=\frac{n+m}{\tau}u+\frac{2}{\tau}u\ln u$.
This follows from $u\in W^{1,2}_{1-\frac{n}{2}}(M)\subset
W^{0,\frac{2n}{n-2}}_{1-\frac{n}{2}}(M)=L^{\frac{2n}{n-2}}(M)$ and the fact
that there exits a constant $a(n)$ such that $|u\ln u|\leq
a(n)+|u|^{\frac{n}{n-2}}$. Indeed,
$\int_{M}\chi^{2\left(-\frac{n}{2}\right)+n}|u\ln u|^{2}\leq a(n)^{2}{\rm
Vol}(M)+\int_{M}|u|^{\frac{2n}{n-2}}<\infty$. So $u\ln u\in
W^{0,2}_{-\frac{n}{2}}\subset W^{0,2}_{1-\frac{n}{2}-2}$.

Then the Nash-Moser iteration as in the proof of Proposition 5.1 gives the
asymptotic behavior in Theorem 1.6.

##  7\. Horn singularity

In this section, we study the spectrum and eigenfunctions of the Schrödinger
operators $-\Delta_{g}+cR_{g}(c\geq 0)$ on compact manifolds with a kind of
more general singularities, whose model is $((0,1)\times
F^{f},g_{\gamma}=dr^{2}+r^{2\gamma}\hat{g})$. Here $\gamma>0$, and $\hat{g}$
is a Riemannian metric on $F^{f}$. This is a $r^{\gamma}$-horn in [Che80]. If
in particular $\gamma=1$, then this is a model cone singularity that we
studied in preceding sections. So now we assume $\gamma>1$, and this kind of
singularities appear in the study of singular projective varieties.

The model singular metric $g_{\gamma}=dr^{2}+r^{\gamma}\hat{g}$ has Laplacian
$\Delta_{g_{\alpha}}$ and scalar curvature $R_{g}$ as

$\displaystyle\Delta_{g_{\gamma}}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}}{\partial
r^{2}}+\frac{\gamma(n-1)}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2\gamma}}\Delta_{\hat{g}},$ $\displaystyle R_{g_{\gamma}}$
$\displaystyle=$
$\displaystyle\frac{R_{\hat{g}}}{r^{2\alpha}}-\frac{\gamma(n-1)(n\gamma-2)}{r^{2}},$

where $n=f+1$.

###  7.1. manifolds with $r^{\gamma}$-horn singularities

Similar as Definition 2.1, we define

######  Definition 7.1.

We say that $(M^{n},g)$ is a compact Riemannian manifold with
$r^{\gamma}$-horn singularities ($\gamma>1$), if

  1. (1)

$M^{n}$ is a compact $n$-dimensional manifold with boundary $\partial M$.

  2. (2)

The boundary $\partial M$ endowed with a Riemannian metric $g_{\partial M}$ is
the total space of a Riemannian submersion

(7.1) $\pi:(\partial M,g_{\partial M})\longrightarrow(B^{b},\check{g})$

with a $b$-dimensional closed Riemannian manifold $(B^{b},\check{g})$ as base
and a $f$-dimensional closed manifold $F^{f}$ as a typical fiber, in
particular, $b+f=n-1$, and

(7.2) $g_{\partial M}=\hat{g}+\pi^{*}\check{g},$

where $\hat{g}$ denotes metrics on fibers induced by $g_{\partial M}$;

  3. (3)

Moreover, there exists a collar neighborhood $U=(0,1)\times\partial M$ of the
boundary $\partial M$ in $M$ with the radial coordinate $r$ $(r=0$ corresponds
to the boundary $\partial M)$, and the metric $g$ restricted on $U$ satisfies
$g=g_{\gamma}+h$ as $r\rightarrow 0$, where the Riemannian metric $g_{\gamma}$
on $U$ given by

$g_{\gamma}=dr^{2}+r^{2\gamma}\hat{g}+\pi^{*}\check{g},$

and $h$ satisfies

$|r^{\gamma
k}\nabla^{k}_{g_{\gamma}}h|_{g_{\gamma}}=O(r^{\alpha}),\quad\text{as}\ \
r\rightarrow 0,$

for some $\alpha>0$ and all $k\in\mathbb{N}$.

###  7.2. $\alpha$-weighted Sobolev spaces

Let $(M^{n},g)$ be a compact Riemannian manifold with $\alpha$-horn
singularities as defined in Definition 7.1. For each $p\geq 1$,
$k\in\mathbb{N}$ and $\delta\in\mathbb{R}$, the $\alpha$-weighted Sobolev
space $W^{k,p}_{\delta,\alpha}(M)$ is the completion of
$C^{\infty}_{0}(\mathring{M})$ with respect to the $\gamma$-weighted Sobolev
norm

(7.3)
$\|u\|_{W^{k,p}_{\delta,\gamma}(M)}=\left(\int_{M}\left(\sum^{k}_{i=0}\chi^{p(\delta-\gamma
i)+n\gamma}|\nabla^{i}u|^{p}_{g}\right)d{\rm vol}_{g}\right)^{\frac{1}{p}},$

where $\nabla^{i}u$ denotes the $i$-times covariant derivative of the function
$u$, and $\chi\in C^{\infty}(\mathring{M})$ is a positive weight function
satisfying

(7.4) $\chi(x)=\begin{cases}1,&\text{if}\ \ x\in M\setminus U,\\\
\frac{1}{r},&\text{if}\ \ r=dist(x,\partial M)<\frac{1}{10},\end{cases}$

and $0<(\chi(x))^{-1}\leq 1$ for all $x\in\mathring{M}$. Recall that
$U=(0,1)\times\partial M\subset M$ is the collar neighborhood of the boundary
$\partial M$.

On a manifold $(M^{n},g)$ with isolated $r^{\gamma}$-horn singularities, i.e.
the base manifold $B$ in Definition 7.1 is a point, one can derive the
following weighted Sobolev inequalities.

######  Proposition 7.2.

Let $(M^{n},g)$ be a manifold with an isolated $r^{\gamma}$-horn singularity.
For any $\delta\in\mathbb{R}$, $1\leq p<n$, $0\leq l\leq k$, and $q$ with
$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}>0$, there exists a constant $C$, such
that

$\|u\|_{W^{l,q}_{\delta,\gamma}(M,g)}\leq
C\|u\|_{W^{k,p}_{\delta,\gamma}(M,g)}$

holds for all $u\in C^{\infty}_{0}(\mathring{M})$.

Consequently, there is a continuous embedding

$W^{k,p}_{\delta,\gamma}(M,g)\subset W^{l,q}_{\delta,\gamma}(M,g).$

###### Proof.

The key of establishment of the $\gamma$-weighted Sobolev embedding is to
derive it on a collar neighborhood $(0,1)\times\partial M$ of $\partial M$
with the model metric $g_{\gamma}=dr^{2}+r^{2\gamma}\hat{g}$.

For any $u\in C^{\infty}_{0}((0,1)\times\partial M)$,

$\displaystyle\|u\|_{W^{0,\frac{np}{n-p}}_{\delta,\gamma}((0,1)\times\partial
M,g_{\gamma})}$ $\displaystyle=$ $\displaystyle\left(\int_{(0,1)\times\partial
M}(u(r,x))^{\frac{pn}{n-p}}\left(\frac{1}{r}\right)^{\frac{pn}{n-p}\delta+n\gamma}d{\rm
vol}_{g_{\gamma}}\right)^{\frac{n-p}{np}}$ $\displaystyle=$
$\displaystyle\left(\int_{\partial
M}\int^{1}_{0}\left(u(r,x)\right)^{\frac{pn}{n-p}}\left(\frac{1}{r}\right)^{\frac{pn}{n-p}\delta+n\gamma}r^{(n-1)\gamma}drd{\rm
vol}_{\hat{g}}\right)^{\frac{n-p}{np}}$ $\displaystyle=$
$\displaystyle\left(\int_{\partial
M}\int^{1}_{0}\left(u(r,x)r^{-\delta}\right)^{\frac{pn}{n-p}}r^{-\gamma}drd{\rm
vol}_{\hat{g}}\right)^{\frac{n-p}{np}}$

In the last integral, we make a change of variable $r=s^{\frac{1}{1-\gamma}}$,
and set $\tilde{u}(s,x)=u(s^{\frac{1}{1-\gamma}},x)$. Then $\tilde{u}\in
C^{\infty}_{0}((1,\infty)\times\partial M)$, and one has

$\displaystyle\|u\|_{W^{0,\frac{np}{n-p}}_{\delta,\gamma}((0,1)\times\partial
M,g_{\gamma})}$ $\displaystyle=$ $\displaystyle\left(\int_{\partial
M}\int^{\infty}_{1}\left(\tilde{u}(s,x)s^{-\frac{\delta}{1-\gamma}}\right)^{\frac{np}{n-p}}\frac{1}{\gamma-1}dsd{\rm
vol}_{\hat{g}}\right)^{\frac{n-p}{np}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\gamma-1}\right)^{\frac{n-p}{np}}\|\tilde{u}(s,x)s^{-\frac{\delta}{1-\gamma}}\|_{L^{\frac{np}{n-p}}((1,\infty)\times\partial
M,g_{0})}.$

Here $g_{0}=ds^{2}+\hat{g}$ is the product metric on $(1,\infty)\times\partial
M$. Then the Sobolev inequality on the cylinder $((1,\infty)\times\partial
M,g_{0})$ produces

$\displaystyle\|u\|_{W^{0,\frac{np}{n-p}}_{\delta,\gamma}((0,1)\times\partial
M,g_{\gamma})}$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{\gamma-1}\right)^{\frac{n-p}{np}}C\|\tilde{u}(s,x)s^{-\frac{\delta}{1-\gamma}}\|_{W^{1,p}((1,\infty)\times\partial
M,g_{0})}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\gamma-1}\right)^{\frac{n-p}{np}}C\left(\int_{\partial
M}\int^{\infty}_{1}\left(|\nabla(\tilde{u}(s,x)s^{-\frac{\delta}{1-\gamma}})|^{p}_{g_{0}}+|\tilde{u}(s,x)|^{p}s^{-\frac{\delta
p}{1-\gamma}}\right)dsd{\rm vol}_{\hat{g}}\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{\gamma-1}\right)^{\frac{n-p}{np}}C\Bigg{(}\int_{\partial
M}\int^{\infty}_{1}\Bigg{(}\left(\frac{\delta}{\gamma-1}\right)^{p}s^{\frac{p\delta}{\gamma-1}-p}|\tilde{u}(s,x)|^{p}+s^{-\frac{\delta
p}{1-\gamma}}\left|\frac{\partial}{\partial s}\tilde{u}(s,x)\right|^{p}$
$\displaystyle\qquad+s^{-\frac{p\delta}{1-\gamma}}|\nabla^{\hat{g}}\tilde{u}(s,x)|^{p}_{\hat{g}}+s^{-\frac{p\delta}{1-\gamma}}|\tilde{u}(s,x)|^{p}\Bigg{)}dsd{\rm
vol}_{\hat{g}}\Bigg{)}^{\frac{1}{p}}$

Now change the variable back, i.e. let $s=r^{1-\gamma}$, and note
$\frac{\partial\tilde{u}}{\partial s}(s,x)=\frac{\partial u}{\partial
r}(r,x)\left(\frac{1}{1-\gamma}\right)r^{\gamma}$.One has

$\displaystyle\|u\|_{W^{0,\frac{np}{n-p}}_{\delta,\gamma}((0,1)\times\partial
M,g_{\gamma})}$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{\gamma-1}\right)^{\frac{n-p}{np}}C\Bigg{(}\int_{\partial
M}\int^{1}_{0}\Bigg{(}\left(\frac{\delta}{\gamma-1}\right)^{p}r^{-p\delta+p(\gamma-1)}|u(r,x)|^{p}$
$\displaystyle\qquad+r^{-p\delta+p\gamma}\left|\frac{1}{\gamma-1}\right|^{p}\left|\frac{\partial
u}{\partial
r}(r,x)\right|^{p}+r^{-p\delta}|\nabla^{\hat{g}}u(r,x)|^{p}_{\hat{g}}$
$\displaystyle\qquad+r^{-p\delta}|u(r,x)|^{p}\Bigg{)}(\gamma-1)r^{-\alpha}drd{\rm
vol}_{\hat{g}}\Bigg{)}^{\frac{1}{p}}$ $\displaystyle\leq$ $\displaystyle
C\Bigg{(}\int_{\partial
M}\int^{1}_{0}\Bigg{[}\left(\frac{1}{r}\right)^{p(\delta-\gamma)+n\gamma}\left(\left|\frac{\partial
u}{\partial
r}(r,x)\right|^{p}+\left(\frac{1}{r}\right)^{p\gamma}|\nabla^{\hat{g}}u(r,x)|^{p}_{\hat{g}}\right)$
$\displaystyle\qquad+\left(\frac{1}{r}\right)^{p\delta+n\gamma}|u(r,x)|^{p}\Bigg{]}r^{(n-1)\gamma}drd{\rm
vol}_{\hat{g}}\Bigg{)}^{\frac{1}{p}}$ $\displaystyle\leq$ $\displaystyle
C\Bigg{(}\int_{(0,1)\times\partial
M}\Bigg{[}\left(\frac{1}{r}\right)^{p(\delta-\gamma)+n\gamma}|\nabla^{g_{\gamma}}u|^{p}_{g_{\gamma}}+\left(\frac{1}{r}\right)^{p\delta+n\gamma}|u(r,x)|^{p}\Bigg{]}d{\rm
vol}_{g_{\gamma}}\Bigg{)}^{\frac{1}{p}}$ $\displaystyle=$ $\displaystyle
C\|u\|_{W^{1,p}_{\delta,\gamma}((0,1)\times\partial M,g_{\gamma})}.$

Now combining this with the usual Sobolev inequality in the interior part of
the manifold $M$, one can obtain that for any given $\delta\in\mathbb{R}$ and
$p\geq 1$, there exists a constant $C$ such that

(7.5) $\|u\|_{W^{0,\frac{np}{n-p}}_{\delta,\gamma}(M,g)}\leq
C\|u\|_{W^{1,p}_{\delta,\gamma}(M,g)}$

for all $u\in C^{\infty}_{0}(\mathring{M})$.

Then by the inequality (7.5), Kato’s inequality and the definition of the
$\gamma$-weighted Sobolev norm, one has

$\|\nabla u\|_{W^{0,\frac{np}{n-p}}_{\delta,\gamma}(M,g)}\leq C\|\nabla
u\|_{W^{1,p}_{\delta,\gamma}(M,g)}\leq
C\|u\|_{W^{2,p}_{\delta+\gamma,\gamma}(M,g)}.$

Moreover,

$\|u\|_{W^{0,\frac{np}{n-p}}_{\delta+\gamma,\gamma}(M,g)}\leq
C\|u\|_{W^{1,p}_{\delta+\gamma,\gamma}(M,g)}\leq
C\|u\|_{W^{2,p}_{\delta+\gamma,\gamma}(M,g)}.$

Thus

$\|u\|_{W^{1,\frac{np}{n-p}}_{\delta+\gamma,\gamma}(M,g)}\leq
C\|u\|_{W^{2,p}_{\delta+\gamma,\gamma}(M,g)}.$

Because in this inequality $\delta$ is arbitrary, one obtains that for any
given $\delta\in\mathbb{R}$ and $p\geq 1$ there exits a constant $C$ such that

$\|u\|_{W^{1,\frac{np}{n-p}}_{\delta,\gamma}(M,g)}\leq
C\|u\|_{W^{2,p}_{\delta,\gamma}(M,g)}$

for all $u\in C^{\infty}_{0}(\mathring{M})$.

Then one can inductively prove the $\gamma$-weighted Sobolev inequality and
complete the proof of the proposition. ∎

In Proposition 3.4, we established the weighted Sobolev inequality on
manifolds with conical singularities by using Sobolev inequality on these
manifolds. A Hardy’s inequality is the key ingredient in the derivation of the
Sobolev inequality on manifolds with conical singularities. This does not work
for manifolds with $r^{\gamma}$-horn singularities any more. On the other
hand, it is interesting to note that, for projective varieties, the work of P.
Li and G. Tian [LT95] implies the Sobolev inequality via their upper bound for
the heat kernel.

In the isolated $r^{\gamma}$-horn singularities case, we have obtained the
$\gamma$-weighted Sobolev inequalities. In the non-isolated singularities
case, if we assume that the Sobolev inequalities holds, e.g. on projective
varieties with $r^{\gamma}$-horn singularities, then similarly as in the proof
of Proposition 3.4, we can derive $\gamma$-weighted Sobolev inequalities.

######  Proposition 7.3.

Let $(M^{n},g)$ be a compact Riemannian manifold with non-isolated
$r^{\gamma}$-horn singularities, on which the Sobolev inequalities hold. For
any $\delta\in\mathbb{R}$, $1\leq p<n$, $0\leq l\leq k$, and $q$ with
$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}>0$, there exists a constant $C$, such
that

$\|u\|_{W^{l,q}_{\delta,\gamma}}\leq C\|u\|_{W^{k,p}_{\delta,\gamma}}.$

holds for all $u\in C^{\infty}_{0}(\mathring{M})$.

Consequently, there is a continuous embedding

$W^{k,p}_{\delta,\gamma}(M)\subset W^{l,q}_{\delta,\gamma}(M).$

Moreover, we have the following compact $\alpha$-weighted Sobolev embedding,
which will play an important role in the study of the spectrum of the
Schödinger operator $\Delta+cR$.

######  Proposition 7.4.

Let $(M,g)$ be a compact Riemannian manifold with $r^{\gamma}$-horn
singularities. The continuous embedding
$W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M,g)\hookrightarrow L^{2}(M,g)$ is
compact.

######  Remark 7.5.

Note that the embedding
$W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M,g)\hookrightarrow L^{2}(M,g)$
does not follow from Proposition 7.3, since in Proposition 7.4 we do not
assume that the Sobolev inequalities hold on the manifold with
$r^{\gamma}$-horn singularities. In the following proof, we will obatin both
the embedding $W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M,g)\hookrightarrow
L^{2}(M,g)$ and its compactness simultaneously.

###### Proof.

Clearly, it suffices to obtain the compact embedding on a collar neighborhood
$U=(0,1)\times\partial M$ of $\partial M$ with the model metric
$g_{\alpha}=dr^{2}+r^{2\gamma}\hat{g}+\pi^{*}\check{g}$. Similarly as in the
proof of Lemma 3.7, we relate the function spaces on the model collar
neighborhood $(U,g_{\gamma})$ to that on a finite cylinder $(C(\partial
M)=(0,1)\times\partial M,g_{C}=dr^{2}+g_{\partial M})$.

The mapping

$\displaystyle L^{2}(U,g_{\gamma})$ $\displaystyle\rightarrow$ $\displaystyle
L^{2}(C(\partial M),g_{C})$ $\displaystyle u$ $\displaystyle\mapsto$
$\displaystyle\tilde{u}=r^{\frac{\gamma f}{2}}u$

is unitary. Similarly as the proof of the inequality in $(\ref{eqn: cone
cylinder inequality})$ in Lemma 3.7, we can show that

(7.6)
$\|u\|_{W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(U,g_{\gamma})}\geq\|u\|_{W^{1,2}(C(\partial
M),g_{C})}.$

Then by combining the inequality in $(\ref{eqn: horn cylinder inequality})$
with the classic Rellich lemma on a finite cylinder, we complete the proof. ∎

Finally, we derive the following compact $\alpha$-weighted Sobolev embedding
that will be used in the study of $W$-functional on manifolds with
$r^{\gamma}$-horn singularities.

######  Proposition 7.6.

Let $(M^{n},g)$ be a compact Riemannian manifold with $r^{\gamma}$-horn
singularities. There is a compact embedding
$W^{1,1}_{\gamma-n\gamma,\gamma}(M,g)\subset L^{1}(M,g)$.

###### Proof.

Again, it suffices to derive the compact embedding on a small collar
neighborhood of the singular set $\partial M$ with the model metric
$g_{\gamma}$. Because $\gamma>1$, there exists $\epsilon>0$ such that for all
$r<\epsilon$

$r^{-\gamma}-\gamma fr^{-1}>1.$

The map

$\displaystyle L^{1}((0,\epsilon)\times\partial M,g_{\gamma})$
$\displaystyle\rightarrow$ $\displaystyle L^{1}((0,\epsilon)\times\partial
M,g_{Cyl})$ $\displaystyle u$ $\displaystyle\mapsto$
$\displaystyle\tilde{u}=r^{\gamma f}u$

is unitary. Here $g_{Cyl}$ is the product metric $dr^{2}+g_{\partial M}$.

On the other hand,

$\displaystyle\|u\|_{W^{1,1}_{\gamma-n\gamma,\gamma}((0,\epsilon)\times\partial
M,g_{\gamma})}$ $\displaystyle=$ $\displaystyle\int_{\partial
M}\int^{\epsilon}_{0}\left(|\nabla
u|_{g_{\gamma}}+\frac{1}{r^{\gamma}}|u|\right)r^{\gamma f}drd{\rm
vol}_{g_{\partial M}}$ $\displaystyle\geq$ $\displaystyle\int_{\partial
M}\int^{\epsilon}_{0}\left(|\partial_{r}u|+|\nabla u|_{g_{\partial
M}}+\frac{1}{r^{\gamma}}|u|\right)r^{\gamma f}drd{\rm vol}_{g_{\partial M}}$
$\displaystyle\geq$ $\displaystyle\int_{\partial
M}\int^{\epsilon}_{0}\left(|\partial_{r}\tilde{u}|+|\nabla\tilde{u}|_{g_{\partial
M}}+(r^{-\gamma}-\gamma fr^{-1})|\tilde{u}|\right)drd{\rm vol}_{g_{\partial
M}}$ $\displaystyle\geq$
$\displaystyle\|\tilde{u}\|_{W^{1,1}((0,\epsilon)\times\partial M,g_{Cyl})}.$

Then Rellich lemma on the cylinder $((0,\epsilon)\times\partial M,g_{Cyl})$
completes the proof. ∎

###  7.3. Spectrum of $-\Delta_{g}+cR_{g}$ on manifolds with $r^{\gamma}$-horn
singularities

######  Theorem 7.7.

Let $(M,g)$ be a compact Riemannian manifold with $r^{\gamma}$-horn
singularities as in Definition 7.1 with $\gamma>1$. If $\min\limits_{\partial
M}\\{R_{\hat{g}}\\}>0$, then the Friedrichs extension of the Schrödinger
operator $-\Delta_{g}+cR_{g}(c>0)$ with the domain
$C^{\infty}_{0}(\mathring{M})$ has the spectrum consisting of discrete
eigenvalues with finite multiplicity
$-\infty<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\cdots$, and the
corresponding eigenfunctions are smooth and form a basis of $L^{2}(M,g)$.

###### Proof.

The scalar curvature of the metric $g$ restricted on a small collar
neighborhood of $\partial M$ is

(7.7)
$R_{g}=\frac{R_{\hat{g}}}{r^{2\gamma}}-\frac{f\gamma(n\gamma+2)}{r^{2}}+R_{\check{g}}\circ\pi-|A|^{2}-|T|^{2}-|N|^{2}-\check{\delta}N+o(r^{-2\gamma+\alpha}).$

Thus for sufficiently small $\epsilon>0$, on $(0,\epsilon)\times\partial M$,
one has

$R_{g}\geq\frac{\delta}{r^{2\gamma}},$

for some $\delta>0$. Consequently, for any $u\in
C^{\infty}_{0}((0,\epsilon)\times\partial M)$,

$\begin{split}\langle(-\Delta_{g}+cR_{g})u,u\rangle_{L^{2}(M,g)}&=\int_{M}(-\Delta_{g}u+cR_{g}u)ud{\rm
vol}_{g}\\\ &=\int_{M}|\nabla u|^{2}_{g}+cR_{g}u^{2}d{\rm vol}_{g}\\\
&\geq\int_{M}|\nabla u|^{2}_{g}+\frac{c\delta}{r^{2\gamma}}u^{2}d{\rm
vol}_{g}\\\
&\geq\min\\{1,c\delta\\}\|u\|_{W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M,g)}.\end{split}$

Then by choosing a sufficiently large constant $A$ to control the scalar
curvature $R_{g}$ on the interior part $M\setminus(0,\epsilon/2)\times\partial
M$, one can obtain that there exists a constant $C$ such that

$\langle(-\Delta_{g}+cR_{g}+A)u,u\rangle_{L^{2}(M,g)}\geq
C\|u\|_{W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M,g)},$

for any $u\in C^{\infty}_{0}(\mathring{M})$. So the operator
$-\Delta_{g}+cR_{g}+A$ with domain $C^{\infty}_{0}(\mathring{M})$ has a
Friedrichs self-adjoint extension with the domain contained in
$W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M,g)$ and the image equal to
$L^{2}(M,g)$. Now the compactness of the embedding in Proposition 7.4 implies
that the inverse of $-\Delta_{g}+cR_{g}+A$ is a compact operator from
$L^{2}(M,g)$ to itself. Then the spectrum theory of self-adjoint compact
operators completes the proof. ∎

###  7.4. Perelman’s $\lambda$-functional and $W$-functional on manifolds with
$r^{\gamma}$-horn singularities

By Theorem 7.7, one can define the $\lambda$-functional on a manifold
$(M^{n},g)$ with $r^{\gamma}$-horn singularities as the first eigenvalue of
the Friedrichs extension of the operator $-4\Delta_{g}+R_{g}$, or equivalently

$\lambda(g):=\inf\left\\{\int_{M}\left(4|\nabla
u|^{2}_{g}+R_{g}u^{2}\right)d{\rm vol}_{g}\mid u\in
W^{1,2}_{\gamma-\frac{n\gamma}{2}},\quad\|u\|_{L^{2}(M,g)}=1\right\\}>-\infty,$

provided $\min\limits_{\partial M}\\{R_{\hat{g}}\\}>0$.

For the $W$-functional, we can use $\gamma$-weighted Sobolev inequality in
Proposition 7.3 to control $\int_{M}u^{2}\ln ud{\rm vol}_{g}$ term as follows.
For a fixed $\tau>0$ and any $u\in W^{1,2}_{\gamma-\frac{n\gamma}{2}}(M)$ with
$\|u\|_{L^{2}}=(4\pi\tau)^{\frac{n}{2}}$, one has

$\displaystyle\int_{M}u^{2}\ln ud{\rm vol}_{g}$ $\displaystyle\leq$
$\displaystyle c(n)\int_{M}u^{2+\frac{2}{n}}d{\rm vol}_{g}$
$\displaystyle\leq$ $\displaystyle
c(n)\epsilon\int_{M}u^{2+\frac{4}{n}}+c(n)\frac{1}{\epsilon}\int_{M}u^{2}d{\rm
vol}_{g}$ $\displaystyle\leq$ $\displaystyle
c(n)\epsilon\left(\int_{M}u^{\frac{2n}{n-2}}d{\rm
vol}_{g}\right)^{\frac{n-2}{n}}\cdot\left(\int_{M}u^{2}d{\rm
vol}_{g}\right)^{\frac{2}{n}}+c(n)\frac{1}{\epsilon}(4\pi\tau)^{n}$
$\displaystyle=$ $\displaystyle
c(n)\epsilon(4\pi\tau)^{2}\left(\int_{M}u^{\frac{2n}{n-2}}d{\rm
vol}_{g}\right)^{\frac{n-2}{n}}+c(n)\frac{1}{\epsilon}(4\pi\tau)^{n}$
$\displaystyle\leq$ $\displaystyle
c(n)\epsilon(4\pi\tau)^{2}C\int_{M}\left(|\nabla
u|^{2}_{g}+\chi^{2\gamma}u^{2}d{\rm vol}_{g}\right)d{\rm
vol}_{g}+c(n)\frac{1}{\epsilon}(4\pi\tau)^{n}.$

Here $c(n)$ is a constant depending only on $n$. Now we can choose $\epsilon$
sufficiently small so that

$2c(n)\epsilon(4\pi\tau)^{2}C<\tau\min\limits_{\partial M}\\{R_{\hat{g}}\\}.$

Thus one has

$\inf\left\\{\int_{M}[\tau(4|\nabla u|_{g}+R_{g}u^{2})]-2u^{2}\ln ud{\rm
vol}_{g}\mid u\in W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M),\ \
\|u\|_{L^{2}(M)}=(4\pi\tau)^{\frac{n}{2}}\right\\}>-\infty,$

provided $\min\limits_{\partial M}R_{\hat{g}}>0$.

Hence on a compact manifold with $r^{\gamma}$-horn singularities with
$\min\limits_{\partial M}R_{\hat{g}}>0$, one has

$\inf\left\\{W(g,\tau,u)\mid u\in
W^{1,2}_{\gamma-\frac{n\gamma}{2},\gamma}(M),\ \
\|u\|_{L^{2}(M)}=(4\pi\tau)^{\frac{n}{2}}\right\\}>-\infty.$

###  7.5. Asymptotic of eigenfunctions of $-\Delta_{g}+cR_{g}$ and the minizer
of the $W$-functional on manifolds with $r^{\gamma}$-horn singularities

Similar as the derivation of weighted elliptic estimate inequality in the
conical singularities case in Proposition 3.10, one can obtain the
$\gamma$-weighted elliptic estimate inequality. Then combining this with
$\gamma$-weighted Sobolev inequality in Proposition 7.3, Nash-Moser iteration
give the following asymptotic estimate for eigenfunctions of of
$-\Delta_{g}+cR_{g}$ and the minizer of the $W$-functional on manifolds with
$r^{\gamma}$-horn singularities.

######  Proposition 7.8.

Let $u$ be an eigenfunction of $-\Delta_{g}+cR_{g}$ or the minimizer of the
$W$-functional on manifolds with $r^{\gamma}$-horn singularities. Then one has

$|\nabla^{i}u|=o(r^{-\frac{n-2}{2}\gamma-i\gamma})$

as $r\rightarrow 0$, i.e. approaching to the singular set, for $i=0,1$.

##  8\. Partial asymptotic expansion of eigenfunctions on manifolds with
isolated conical singularities

In this section, we derive a partial asymptotic expansion for the
eigenfunctions of the Schrödinger operator $L:=-4\Delta+R$ on manifolds with
isolated conical singularities in Theorem 8.3. The b-calculus theory of
Melrose is the main tool in this derivation. So in §8.1 we recall some basics
of the b-calculus theory.

###  8.1. Basics of b-calculus of Melrose

In this subsection, we briefly recall some basic notions and results in
b-calculus theory of Melrose that we need in this paper, and we refer to
Melrose’s book, [Mel93], for details. The b-calculus theory is very useful in
studies of elliptic operators on various non-compact manifolds, for example,
complete manifolds with certain ends (e.g. Example 8.1), and manifolds with
conical singularities (Example 8.2).

Let $M^{n}$ be a $n$-dimensional manifold with boundary $\partial M$. Let
$\left\\{y_{1},\cdots,y_{n-1}\right\\}$ be a local coordinate system on
$\partial M$. Combining this with a defining function $x$ of the boundary,
that is $x:\overline{M}\rightarrow[0,+\infty)$ smooth with $x^{-1}(0)=\partial
M$ and $\nabla x\not=0$ on $\partial M$, we obtain a local coordinate system
on $M$.

The fundamental object in the b-calculus theory is the space of vector fields
tangent to the boundary at each point on the boundary. That is denoted by
$\mathcal{V}_{b}$ and can be expressed as

(8.1) $\mathcal{V}_{b}=\left\\{X\in C^{\infty}(M,TM)\ \ \mid\ \
X_{\upharpoonright{\partial M}}\in C^{\infty}(\partial M,T\partial
M)\right\\},$

where $TM$ and $T\partial M$ are tangent bundles of $M$ and $\partial M$,
respectively, and $C^{\infty}(M,TM)$ and $C^{\infty}(\partial M,T\partial M)$
are their smooth sections. Then a vector field $X\in\mathcal{V}_{b}$ can be
expressed locally as:

(8.2)
$X=X_{0}x\partial_{x}+\sum^{n-1}_{i=1}X_{i}\partial_{y_{i}},\quad\text{where}\
\ X_{0},X_{i}\in C^{\infty}(\overline{M}).$

An important point here is that $\mathcal{V}_{b}$ can be realized as the space
of all smooth sections of a vector bundle, which is denoted by ${}^{b}TM$ and
called $b$-tangent bundle, i.e.

(8.3) $\mathcal{V}_{b}=C^{\infty}(M,{{}^{b}T}M).$

Then the dual bundle of ${}^{b}TM$ is denoted by ${}^{b}T^{*}M$, whose smooth
sections $\omega\in C^{\infty}(M,{{}^{b}T}^{*}M)$ can be expressed locally as:

(8.4)
$\omega=\omega_{0}\frac{dx}{x}+\sum^{n-1}_{i=1}\omega_{i}dy_{i},\quad\text{where}\
\ \omega_{0},\omega_{i}\in C^{\infty}(\overline{M}).$

A $b$-differential operator of order $k$ is defined to be a linear map
$P:C^{\infty}(M)\to C^{\infty}(M)$ given by a finite sum of up to $k$-fold
products of elements of $\mathcal{V}_{b}$ and $C^{\infty}(M)$, i.e.

(8.5) $P=\sum_{j\in J,\ \ l(j)\leq k}a_{j}X_{1,j}\cdots
X_{l(j),j},\quad\text{where}\ \ a_{j}\in C^{\infty}(M),\ \
X_{i},j\in\mathcal{V}_{b}$

The set of all $b$-differential operators of order $k$ on $M$ is denoted by
${\rm Diff}^{k}_{b}(M)$. Then $P\in{\rm Diff}^{k}_{b}(M)$ if and if only $P$
is a differential operator of order $k$ (in the usual sense), and near
$\partial M$ it can be expressed as

(8.6) $P=\sum_{|\alpha|\leq k}a_{\alpha}(x,y_{1},\cdots,y_{n-1})(x\partial
x)^{\alpha_{0}}(\partial_{y_{1}})^{\alpha_{1}}\cdots(\partial_{y_{n-1}})^{\alpha_{n-1}},$

where
$\alpha=(\alpha_{0},\alpha_{1},\cdots,\alpha_{n-1})\in\mathbb{N}_{0}^{n}$ are
multi-indices, and the coefficients $a_{\alpha}$ are smooth up to $\partial
M$.

For $P\in{\rm Diff}^{k}_{b}(M)$ , the principal symbol of $P$, denoted by
${{}^{b}\sigma}_{k}(P)$, is a smooth section of the bundle of $k$-symmetric
powers $\bigodot^{k}\left({{}^{b}TM}\right)$. It can be viewed as a homogenous
polynomial of degree $k$ on ${}^{b}T^{*}M$, and if $P$ is locally expressed as
$(\ref{eqn: b-operator local expression})$ near $\partial M$, then
${{}^{b}\sigma}_{k}(P)$ is locally given by

(8.7)
${{}^{b}\sigma}_{k}(P)(x,y,\omega)=\sum_{|\alpha|=k}a_{\alpha}(x,y_{1},\cdots,y_{n-1})(\omega_{0})^{\alpha_{0}}(\omega_{1})^{\alpha_{1}}\cdots(\omega_{n-1})_{\alpha_{n-1}},$

for
$\omega=\omega_{0}(x,y)x\partial_{x}+\sum\limits^{n-1}_{i=1}\omega_{i}(x,y)\partial_{y_{i}}\in{{}^{b}T}^{*}M$.
Then $P\in{\rm Diff}^{k}_{b}(M)$ is said to be $b$-elliptic if
${{}^{b}\sigma}_{k}(P)\neq 0$ on ${{}^{b}T}^{*}M\setminus 0$.

In order to extend the parametrix construction of elliptic operators on
compact smooth manifolds without boundary to b-elliptic operators $P$, it is
natural to consider weighted Sobolev spaces $W^{m,2}_{\delta}(M)$ to control
asymptotic behavior of functions near boundary. But the map:

(8.8) $P:W^{m,2}_{\delta}(M)\rightarrow W^{m-k,2}_{\delta}(M)$

may not have a good pararmetrix for some $\delta$ to get Fredholm property. To
find all indices $\delta$ such that the b-elliptic operator $P$ in $(\ref{eqn:
b-map between weighted spaces})$ has a nice parametrix, one needs to
investigate the so called indicial operator of $P$. In the b-calculus theory,
the weighted Sobolev spaces $W^{k,2}_{\delta}(M)$ are usually defined to be

(8.9) $W^{k,2}_{\delta}(M):=x^{\delta}W^{k,2}_{b}(M)=\\{u=x^{\delta}v\ \ \mid\
\ v\in W^{k,2}_{b}(M)\\},$

where $W^{k,2}_{b}(M)$ is the usual Sobolev space on $M$ with respect to the
b-metric, $g$, which near $\partial M$ is

(8.10) $g=\frac{dx^{2}}{x^{2}}+g^{\partial M}$

with $g^{\partial M}$ a smooth metric on $\partial M$. One can check that
these weighted Sobolev spaces are the same as that defined on manifolds with a
single conical singularity in [DW18] (also see §$\ref{subsect: weighted
Sobolev spaces})$, once we view a cone metric as a conformal change of a
b-metric as discussed in Example 8.2

For an operator $P\in{\rm Diff}^{k}_{b}(M)$ locally given by (8.6), its
indicial operator, denoted by $I(P)$, is an (${\mathbb{R}}^{+}$-invariant)
operator on $\partial M\times\mathbb{R}^{+}$,

(8.11) $I(P)=\sum_{|\alpha|\leq
k}a_{\alpha}(0,y_{1},\cdots,y_{n-1})(t\partial_{t})^{\alpha_{0}}(\partial_{y_{1}})^{\alpha_{1}}\cdots(\partial_{y_{n-1}})^{\alpha_{n-1}}.$

It can be identified, via the Mellin transform on $\mathbb{R}^{+}$, with a
family of differential operator on $\partial M$ parametrized by
$z\in\mathbb{C}$, called the indicial family, given by

(8.12) $I(P)_{M}(z):=\sum_{|\alpha|\leq
m}a_{\alpha}(0,y_{1},\cdots,y_{n-1})z^{\alpha_{0}}(\partial_{y_{1}})^{\alpha_{1}}\cdots(\partial_{y_{n-1}})^{\alpha_{n-1}}.$

The set

(8.13) $spec_{b}(P):=\left\\{z\in\mathbb{C}\ \ \mid\ \ I(P)_{M}(z)\text{ is
not invertible on}\ \ C^{\infty}(\partial M)\right\\}$

is a discrete subset of $\mathbb{C}$. Then a real number $\delta\in\mathbb{R}$
is said to be a indicial root of the b-differential operator $P$, if
$\delta={\rm Re}(z)$ for some $z\in spec_{b}(P)$.

For any $\delta\in\mathbb{R}$ that is not a indicial root of b-elliptic
operator $P$, roughly speaking, with the help of inverting the indicial
operator $I(P)$, a parametrix of the operator

(8.14) $P:W^{m,2}_{\delta}(M)\rightarrow W^{m-k,2}_{\delta}(M)$

can be constructed, denoted by

(8.15) $Q:W^{m-k,2}_{\delta}(M)\rightarrow W^{m,2}_{\delta}(M)$

such that

(8.16) $\begin{cases}Q\circ P=id-R_{1},\cr P\circ Q=id-R_{2},\end{cases}$

where

(8.17) $\begin{cases}R_{1}:&W^{k,2}_{\delta}(M)\rightarrow
W^{\infty,2}_{\delta}(M)\cap\mathcal{A}_{phg}(M),\cr
R_{2}:&W^{m-k,2}_{\delta}(M)\rightarrow
W^{\infty,2}_{\delta}(M)\cap\mathcal{A}_{phg}(M)\end{cases}$

are compact operators. Here
$W^{\infty,2}_{\delta}=\cap^{\infty}_{k=1}W^{k,2}_{\delta}$, and
$\mathcal{A}_{phg}(M)$ denotes the space of smooth functions on $M$ that admit
a full asymptotic expansion, as $x\rightarrow 0$, in terms of power of $x$ and
$\ln x$ with smooth coeffiecients. In particular, the map in $(\ref{eqn:
b-elliptic map})$ is Fredholm, if $P$ is b-elliptic and $\delta$ is not a
indicial root of $P$.

######  Example 8.1 (Manifolds with a cylindrical end).

Let $(M^{n},g)$ be a complete manifold with a cylindrical end, that is, there
exists a relatively compact open submanifold $M_{0}\subset M$ such that its
complement, $M\setminus M_{0}$, with metric $g$ is isometric to a cylinder. In
other words, $M\setminus M_{0}$ is diffeomorphic to $[0,+\infty)\times N$ with
metric

(8.18) $g=ds^{2}+g^{N},$

where $g^{N}$ is a smooth metric on compact manifold $N$.

By doing a change of variable, $x=e^{-s}$, the metric $g$ becomes
$g=\frac{dx^{2}}{x^{2}}+g^{N},$ which is a b-metric in $(\ref{eqn:
b-metric})$, and $M$ can be viewed as the interior of the compact manifold
with boundary with a b-metric and $x$ as a defining function of the boundary.
In this change of variable, Laplace operator on $(M,g)$,
$\Delta=\frac{\partial^{2}}{\partial s^{2}}+\Delta_{g^{N}}$ becomes
$(x\partial_{x})^{2}+\Delta_{g^{N}}$. Clearly, this is a b-elliptic
differential operator, with $I(\Delta)$ identified with
$z^{2}+\Delta_{g^{N}}$. Thus, the indicial roots are the square roots of the
eigenvalues of $\Delta_{g^{N}}$. ∎

######  Example 8.2 (Manifolds with a conical singularity).

Let $(M^{n},g,o)$ be a compact manifold with a single conical singularity,
that is a manifold as defined in Definition 2.1 such that base $B=o$ is a
point and near the singular point $o$ the manifold is isometric to
$(0,1)\times F$ with metric

(8.19) $g_{0}=dr^{2}+r^{2}\hat{g}$

where $\hat{g}$ is a smooth metric on $F$ (To emphasize this is also called an
exact conical singularity). Clearly, this metric is conformal to the b-metric
$\frac{dr^{2}}{r^{2}}+\hat{g}$ with conform factor $r^{2}$.

The regular part of the manifold can be viewed as a compact manifold with
boundary. The elliptic operator, $L:=-4\Delta_{g}+R_{g}$, which we study in
this paper, can be expressed near the singular point $o$ as

(8.20)
$L_{0}:=-4\Delta_{g_{0}}+R_{g_{0}}=-4\left(\partial_{r}^{2}+\frac{n-1}{r}\partial_{r}+\frac{1}{r^{2}}\Delta_{\hat{g}}\right)+\frac{R_{\hat{g}}-(n-1)(n-2)}{r^{2}}.$

Then

(8.21)
$r^{2}L_{0}=-4(r\partial_{r})^{2}-4(n-2)r\partial_{r}-4\Delta_{\hat{g}}+R_{\hat{g}}-(n-1)(n-2).$

Thus, $r^{2}L_{0}$ is a b-elliptic operator, with its indicial family given by

(8.22)
$I(r^{2}L_{0})_{M}(z)=-4z^{2}-4(n-2)z+\left(-4\Delta_{\hat{g}}+R_{\hat{g}}\right)-(n-1)(n-2).$

This is not invertible if and only if $z=\frac{-(n-2)\pm\sqrt{\nu-(n-2)}}{2}$,
where $\nu$ is an eigenvalue of the operator $-4\Delta_{\hat{g}}+R_{\hat{g}}$.
Note that these $z$ are all real numbers, if we assume $R_{\hat{g}}>n-2$. Then
for any

(8.23) $\delta\not\in\left\\{\frac{-(n-2)\pm\sqrt{\nu-(n-2)}}{2}\ \ \mid\ \
\nu\ \ \text{is an eigenvalue of}\ \ -4\Delta_{\hat{g}}+R_{\hat{g}}\right\\}$

and $k\in\mathbb{N}_{0}$, the operator

(8.24) $r^{2}L_{0}:W^{k+2,2}_{\delta}(M)\rightarrow W^{k,2}_{\delta}(M)$

has a parametrix $Q$ such that

(8.25) $\begin{cases}Q\circ r^{2}L_{0}=id-R_{1},\cr r^{2}L_{0}\circ Q=id-
R_{2},\end{cases}$

where

(8.26) $\begin{cases}Q:&W^{k,2}_{\delta}(M,g)\rightarrow
W^{k+2,2}_{\delta}(M,g),\cr R_{1}:&W^{k+2,2}_{\delta}(M,g)\rightarrow
W^{\infty,2}_{\delta}(M,g)\cap\mathcal{A}_{phg}(M),\cr
R_{2}:&W^{k,2}_{\delta}(M,g)\rightarrow
W^{\infty,2}_{\delta}(M,g)\cap\mathcal{A}_{phg}(M)\end{cases}$

are bounded operators, where $\mathcal{A}_{phg}(M)$ denotes the space of
smooth functions on $M$ that admit a full asymptotic expansion, as
$r\rightarrow 0$, in terms of power of $r$ and $\ln r$ with smooth
coeffiecients. ∎

###  8.2. A partial asymptotic expansion of eigenfunctions

Let $(M^{n},g,o)$ be a compact manifold with a single conical singularity at
$o$, as in Definition 2.1 but with the base manifold $B$ being a point $o$.
The singular point $o$ has a neighborhood that is isometric to
$C_{1}(F):=(0,1)\times F$ with metric $g=dr^{2}+r^{2}\hat{g}+h=g_{0}+h$, where
$h$ has asymptotic control as in $(\ref{eqn: metric asymptotic control})$,
i.e.

(8.27) $|r^{k}\nabla^{k}_{g_{0}}h|_{g_{0}}=o(r^{\alpha}),$

as $r\rightarrow 0$, for some $\alpha>0$ and all $k\in\mathbb{N}$. Set
$L:=-4\Delta_{g}+R_{g}$.

Now we take a cut-off function $\psi$ on $(0,1)\times F$ such that

(8.28) $\psi=\begin{cases}0,&\text{on}\left(0,\frac{1}{4}\right)\times F,\cr
1,&\text{on}\left(\frac{3}{4},1\right)\times F,\end{cases}$

and use it to modify the metric $g$ near the singular point by letting
$g_{0}+\psi h$ be the new metric on $(0,1)\times F$ and keeping the metric $g$
on $M\setminus\left((0,1)\times F\right)$ unchanged. This new metric is
exactly conical near the singular point $o$. Let $L_{0}$ denote the
Schrödinger operator $-4\Delta+R$ with respect to the new metric on $M$. Then
as seen in Example 8.2, $r^{2}L_{0}$ is a b-elliptic differential operator.
Here we can make a smooth extension of the radial variable $r$ on $(0,1)\times
F$ to the whole manifold $M$.

By the asymptotic control (8.27) of the perturbation term $h$ of the metric
$g$, the difference

(8.29) $L-L_{0}:W^{k+2,2}_{\delta}\rightarrow W^{k,2}_{\delta-2+\alpha}$

is a bounded second order differential operator for any fixed $k\in\mathbb{N}$
and $\delta\in\mathbb{R}$. Here and in the rest of the paper, we will use
$W^{k,p}_{\delta}$ to denote $W^{k,p}_{\delta}(M,g)$ for simplicity of
notations.

######  Theorem 8.3.

Let $u$ be an eigenfunciton of $-4\Delta_{g}+R_{g}$ on a $n$-dimensional
manifold $(M^{n},g)$ with a conical singularity, satisfying that the scalar
curvature of the cross section $(F,\hat{g})$ of the model cone:
$R_{\hat{g}}>(n-2)$. Then

(8.30) $u=a(y)r^{\mu}+o(r^{\mu^{\prime}}),$

as $r\rightarrow 0$ (approaching the singular point), where
$\mu=-\frac{n-2}{2}+\frac{\sqrt{\nu-(n-2)}}{2}$ and $a(y),y\in F,$ is an
eigenfunction of $-4\Delta_{\hat{g}}+R_{\hat{g}}$ with eigenvalue $\nu$ on
$(F,\hat{g})$, and $\mu^{\prime}>\mu$.

Moreover,

(8.31) $|\nabla^{i}(u-a(y)r^{\mu})|=o(r^{\mu^{\prime}-i})$

as $r\rightarrow 0$, for all $i\in\mathbb{N}$.

###### Proof.

The eigenfunction $u$ satisfies the equation: $Lu=\lambda u$, and $u\in
W^{1,2}_{1-\frac{n}{2}}$ by the estimate in Proposition 4.2. Then by using
weighted elliptic estimate in Proposition 3.10, a weighted elliptic
bootstrapping argument gives:

(8.32) $u\in W^{k,2}_{1-\frac{n}{2}},\quad\forall k\in\mathbb{N}.$

The eigenfunction equation $Lu=\lambda u$ can be rewritten as

(8.33) $r^{2}L_{0}u=\lambda r^{2}u+r^{2}(L_{0}-L)u.$

Clearly, the eigenvalues of $-4\Delta_{\hat{g}}+R_{\hat{g}}$: $\nu>(n-2)$,
since $R_{\hat{g}}>(n-2)$. In particular, this implies that
$\delta=1-\frac{n}{2}$ is not a indicial root of the operator $r^{2}L_{0}$ by
$(\ref{eqn: indicial root})$. Applying the parametrix $Q$ in Example 8.2 for
the operator: $r^{2}L_{0}:W^{k,2}_{1-\frac{n}{2}}\rightarrow
W^{k-2,2}_{1-\frac{n}{2}}$, to the equation $(\ref{eqn: rearranged eigen
equation})$ and using (8.25), one obtains a decomposition of $u$ as

(8.34) $u=R_{1}u+Q(\lambda r^{2}u)+Q(r^{2}(L_{0}-L)u).$

By the regularity of $u$ in $(\ref{eqn: eigenfunction regularity})$, we have
$r^{2}u\in W^{k,2}_{3-\frac{n}{2}}$ and so

(8.35) $Q(r^{2}u)\in W^{k,2}_{3-\frac{n}{2}},\quad\forall k\in\mathbb{N}.$

Moreover, by (8.29), we have $r^{2}(L_{0}-L)u\in
W^{k,2}_{1-\frac{n}{2}+\alpha},$ and so

(8.36) $Q(r^{2}(L_{0}-L)u)\in W^{k,2}_{1-\frac{n}{2}+\alpha},\quad\forall
k\in\mathbb{N}.$

Furthermore, $R_{1}u\in W^{\infty,2}_{1-\frac{n}{2}}\cap\mathcal{A}_{phg}$; in
particular, by looking at the first two leading order terms in the full
asymptotic expansion of $R_{1}u$, we have

(8.37) $R_{1}u\sim a_{p}(y)r^{\mu}(\ln r)^{p}+a_{p-1}(y)r^{\mu}(\ln r)^{p-1},$

as $r\rightarrow 0$ for some $\mu\in\mathbb{R}$, $p\in\mathbb{N}\cup\\{0\\}$,
nonzero smooth function $a_{p}(y)$ on $F$, and a smooth function $a_{p-1}(y)$
on $F$ ($a_{p-1}(y)$ may be zero). Hence (up to higher order terms),

$\displaystyle r^{2}L_{0}\circ R_{1}u\sim\\{-\nu
a_{p}(y)+(-4\Delta_{\hat{g}}+R_{\hat{g}})a_{p}(y)\\}r^{\mu}(\ln r)^{p}$ (8.38)
$\displaystyle\quad+\left\\{\sigma a_{p}(y)-\nu
a_{p-1}(y)+\left(-4\Delta_{\hat{g}}+R_{\hat{g}}\right)a_{p-1}(y)\right\\}r^{\mu}(\ln
r)^{p-1},$

as $r\rightarrow 0$, where

(8.39) $\nu:=-[-4\mu(n+\mu-2)-(n-1)(n-2)],\ \ \ \sigma:=-4p(2\mu-1)-4(n-1)p$

On the other hand, by applying $r^{2}L_{0}$ to the equation (8.34), and using
eigenfunction equation and (8.25), we obtain (assuming without loss of
generality that $0<\alpha<2$)

(8.40) $r^{2}L_{0}\circ R_{1}u=\lambda R_{2}(r^{2}u)+R_{2}(r^{2}(L_{0}-L)u)\in
W^{\infty,2}_{1-\frac{n}{2}+\alpha}.$

Claim:

(8.41) $-\nu a_{p}(y)+(-4\Delta_{\hat{g}}+R_{\hat{g}})a_{p}(y)\equiv 0,$

and

(8.42) $\sigma a_{p}(y)-\nu
a_{p-1}(y)+\left(-4\Delta_{\hat{g}}+R_{\hat{g}}\right)a_{p-1}(y)\equiv 0.$

Otherwise, by the asymptotic behavior in $(\ref{eqn: asymptotic r2L0R1u})$,
the inclusion in $(\ref{eqn: r2L0 acts on remainder})$, and the observation in
$(\ref{eqn: polynomial weighted Sobolev condition})$, we obtain

(8.43) $\mu>1-\frac{n}{2}+\alpha.$

Here note that $r^{\mu}(\ln r)^{p},r^{\mu}(\ln r)^{p-1},\cdots$ and $r^{\mu}$
are linearly independent functions. By $(\ref{eqn: asymptotic R1u})$, and
again $(\ref{eqn: polynomial weighted Sobolev condition})$, $(\ref{eqn: mu-
inequality})$ then implies:

(8.44) $R_{1}u\in W^{\infty,2}_{1-\frac{n}{2}+\alpha}.$

Hence by (8.34), (8.35), (8.36) and (8.44), we obtain

(8.45) $u\in W^{\infty,2}_{1-\frac{n}{2}+\alpha}.$

Iterating the above process, we obtain

(8.46) $u\in W^{\infty,2}_{1-\frac{n}{2}+k\alpha},\ \ \forall k\in\mathbb{N}.$

As a result, by Lemma 8.1 in [DW18],

(8.47) $u=o(r^{1-\frac{n}{2}+k\alpha}),\ \ \forall k\in\mathbb{N}.$

The argument also implies

(8.48) $Q(\lambda r^{2}u)+Q(r^{2}(L_{0}-L)u)=o(r^{1-\frac{n}{2}+k\alpha}),\ \
\forall k\in\mathbb{N}.$

Therefore, by (8.34), we have

(8.49) $R_{1}u=o(r^{1-\frac{n}{2}+k\alpha}),\ \ \forall\in\mathbb{N}.$

Then the asymptotic expansion (8.38) implies $a_{p}(y)\equiv 0$, and this
contradicts with the assumption: $a_{p}(y)\not\equiv 0$. Thus Claims
$(\ref{eqn: claim})$ and $(\ref{eqn: claim2})$ hold.

Clearly, the equation $(\ref{eqn: claim})$ says that $a_{p}(y)$ is an
eigenfunction of $-4\Delta_{\hat{g}}+R_{\hat{g}}$ with eigenvalue $\nu$. By
solving (8.39), we obtain

(8.50) $\mu=-\frac{n-2}{2}+\frac{\sqrt{\nu-(n-2)}}{2}.$

Note that we have used our previous result, Theorem 1.4 of [DW18], together
with $\nu>n-2$ via our assumption $R_{\hat{g}}>n-2$, to rule out the negative
sign in the quadratic formula here.

We can write the smooth function $a_{p-1}(y)$ on $F$ as a linear combination
of eigenfunctions of $-4\Delta_{\hat{g}}+R_{\hat{g}}$. Note that
eigenfunctions with eigenvalue $\nu$ will go away in the expression: $-\nu
a_{p-1}(x)+(-4\Delta_{\hat{g}}+R_{\hat{g}})a_{p-1}(x)$, and recall that
$a_{p}(x)$ is an eigenfunction with eigenvalue $\nu$. Thus $a_{p}(y)$ and
$-\nu a_{p-1}(y)+(-4\Delta_{\hat{g}}+R_{\hat{g}})a_{p-1}(y)$ are linearly
independent functions. Thus, $(\ref{eqn: claim2})$ implies:

(8.51) $\sigma=-4p(2\mu-1)-4(n-1)p=0.$

Then by plugging $(\ref{eqn: mu-nu-relation})$ into $(\ref{eqn: p-mu-
relation})$, we obtain

$-4p\sqrt{\nu-(n-2)}=0.$

Hence $p=0$, since $\nu>(n-2)$ (by our assumption $R_{\hat{g}}>(n-2)$).

Now to complete the proof, by the decomposition in (8.34), it suffices to show
that

(8.52) $Q(\lambda r^{2}u)+Q(r^{2}(L_{0}-L)u)=o(r^{\mu^{\prime}}),\ \ \text{for
some}\ \ \mu^{\prime}>\mu.$

By $(\ref{eqn: Q first term})$ and $(\ref{eqn: Q second term})$, we have that

(8.53) $Q(\lambda r^{2}u)+Q(r^{2}(L_{0}-L)u)\in
W^{\infty,2}_{1-\frac{n}{2}+\alpha},$

and so

(8.54) $Q(\lambda r^{2}u)+Q(r^{2}(L_{0}-L)u)=o(r^{1-\frac{n}{2}+\alpha}),$

by Lemma 8.1 in [DW18]. If $1-\frac{n}{2}+\alpha>\mu$, then set
$\mu^{\prime}=1-\frac{n}{2}+\alpha$, and we are done.

Otherwise, $R_{1}u\sim a_{0}(y)r^{\mu}\in W^{\infty,2}_{1-\frac{n}{2}+\beta}$
for any $\beta<\alpha$ by $(\ref{eqn: polynomial weighted Sobolev
condition})$. Fix $0<\beta<\alpha$. Then (8.34) implies $u\in
W^{\infty,2}_{1-\frac{n}{2}+\beta}$. Starting with this inclusion, by
repeating above derivation, one can obtain $Q(\lambda
r^{2}u)+Q(r^{2}(L_{0}-L)u)=o(r^{1-\frac{n}{2}+\beta+\alpha}).$ Now if
$1-\frac{n}{2}+\beta+\alpha>\mu$, then we are done. Otherwise, we can iterate
above process and obtain (8.52). Actually, we have

(8.55) $|\nabla^{i}(Q(\lambda
r^{2}u)+Q(r^{2}(L_{0}-L)u))|_{g}=o(r^{\mu^{\prime}-i})$

for all $i\in\mathbb{N}$.

Thus, we obtain

(8.56) $u=a_{0}(y)r^{\mu}+u_{1},$

and $|\nabla^{i}u_{1}|=o(r^{\mu^{\prime}-i})$, as $r\rightarrow 0$, for some
$\mu^{\prime}>\mu$ and all $i\in\mathbb{N}\cup\\{0\\}$. ∎

Using a similar but more complicated argument, we now derive a partial
asymptotic expansion for the minimizer $u$ of $W$-functional as in Theorem 8.4
below. Recall that the minimizer $u$ satisfies the equation (see (4.17) in
[DW20])

(8.57) $(-4\Delta+R)u=\frac{2}{\tau}u\ln u+\frac{n+m}{\tau}u,$

where $m$ is the infimum of $W$-functional with the scaling parameter $\tau>0$
as in $(\ref{eqn: mu-functional})$. The main difference between this equation
and that of the eigenfunction as discussed in Theorem 8.3 is the $u\ln u$ term
on the right hand of the Euler-Lagrange equation (8.57). So for the simplicity
of presentation, instead of (8.57), we will consider the equation

(8.58) $(-4\Delta+R)u=u\ln u.$

The basic idea of the proof of Theorem 8.4 is similar to that of Theorem 8.3.
We will still use the parametrix of the Schrödinger operator $L_{0}$ in
Melrose’s b-calculus theory. In the derivation for the partial asymptotic
expansion of eigenfunction in Theorem 8.3, another key ingredient is that one
can obtain $u\in W^{\infty,2}_{1-\frac{n}{2}}$ by using a weighted elliptic
bootstrapping with the help of Proposition 3.10. This helps us estimate the
term $Q(\lambda r^{2}u)+Q(r^{2}(L_{0}-L)u)$ in (8.34).

However, the $u\ln u$ term prevents us from getting the weighted $L^{2}$
estimate for high order derivatives of a solution to $(\ref{eqn: u ln u})$.
Instead, we combine the b-calculus theory and weighted $L^{p}$ elliptic
bootstrapping argument to obtain the following asymptotic estimate. This
improves the asymptotic estimate obtained in [DW20].

######  Theorem 8.4.

Let $u>0$ be a solution of

(8.59) $(-4\Delta_{g}+R_{g})u=u\ln u.$

on a manifold $(M^{n},g,o)$ with a conical singularity at $o$, satisfying
$R_{\hat{g}}>(n-2)$. Then

(8.60) $|\nabla^{i}u|=o(r^{\mu-i})$

as $r\rightarrow 0$ (approaching the singular point), for $i=0,1$ and any
$\mu<-\frac{n-2}{2}+\frac{\sqrt{\nu-(n-2)}}{2}$, where $\nu$ is the smallest
eigenvalue of $-4\Delta_{\hat{g}}+R_{\hat{g}}$ on $(F,\hat{g})$.

###### Proof.

We prove the theorem in two steps. In the first step we show that, for some
real number $\mu^{\prime}$ and any $\mu<\mu^{\prime}$,
$|\nabla^{i}u|=o(r^{\mu-i})$ as $r\to 0$, $i=0,1$. In the second step, we
estimate $\mu^{\prime}$ to complete the proof.

Step 1. The solution $u>0$ satisfies the equation $Lu=u\ln u$. So

(8.61) $r^{2}L_{0}u=r^{2}u\ln u+r^{2}(L_{0}-L)u.$

Applying the parametrix $Q$ of the operator:
$r^{2}L_{0}:W^{2,2}_{1-\frac{n}{2}}\to W^{0,2}_{1-\frac{n}{2}}$, to this
equation and using (8.25), we write $u$ as

(8.62) $u=R_{1}u+Q(r^{2}u\ln u)+Q(r^{2}(L_{0}-L)u).$

Recall that $u\in W^{1,2}_{1-\frac{n}{2}}$ and $u$ is smooth by the local
elliptic regularity. By the weighted Sobolev embedding in Proposition 3.4 in
[DW20], we have

(8.63) $u\in W^{0,p}_{1-\frac{n}{2}},\quad\forall 1\leq p\leq\frac{2n}{n-2}.$

Now, for any $\gamma>0$, there is a constant $a(\gamma)$ such that

(8.64) $|u\ln u|\leq a(\gamma)+|u|^{1+\gamma}.$

It follows that, for any small $\gamma>0$,

(8.65) $u\ln u\in W^{0,p}_{\left(1-\frac{n}{2}\right)(1+\gamma)},\quad\forall
1\leq p\leq\frac{2n}{(n-2)(1+\gamma)}.$

In particular, for $p=2$,

(8.66) $Q(r^{2}u\ln u)\in W^{2,2}_{(1-\frac{n}{2})(1+\gamma)+2}.$

Thus, by taking $\gamma$ sufficiently small, we obtain

(8.67) $Q(r^{2}u\ln u)\in W^{2,2}_{1-\frac{n}{2}+\alpha}.$

In addition, as shown in the proof of Theorem 5.2 in [DW20], by Proposition
5.1 in [DW20], we can obtain

(8.68) $u\in W^{2,p}_{(1-\frac{n}{2})(1+\gamma)},\quad\forall 1\leq
p\leq\frac{2n}{(n-2)(1+\gamma)},\ \ \gamma>0.$

Then by the boundedness of the map in (8.29), we have

(8.69) $r^{2}(L_{0}-L)u\in W^{0,2}_{1-\frac{n}{2}+\alpha},$

and so

(8.70) $Q(r^{2}(L_{0}-L)u)\in W^{2,2}_{1-\frac{n}{2}+\alpha}.$

Thus, by $(\ref{eqn: W Q first term})$ and $(\ref{eqn: W Q second term})$, we
obtain

(8.71) $Q(r^{2}u\ln u)+Q(r^{2}(L_{0}-L)u)\in W^{2,2}_{1-\frac{n}{2}+\alpha}.$

On the other hand, $R_{1}u\in
W^{\infty,2}_{1-\frac{n}{2}}\cap\mathcal{A}_{phg}$; in particular, by looking
at the leading order term,

(8.72) $R_{1}u\sim a(y)r^{\mu^{\prime}}(\ln r)^{p},$

as $r\rightarrow 0$ for some $\mu^{\prime}\in\mathbb{R}$. So $R_{1}u\in
W^{2,2}_{\mu}$ for any $\mu<\mu^{\prime}$.

We first Claim:

(8.73) $u\in W^{2,2}_{\mu},\quad\text{for any}\ \ \mu<\mu^{\prime}.$

If $\mu^{\prime}\leq 1-\frac{n}{2}+\alpha$, then

(8.74) $Q(r^{2}u\ln u)+Q(r^{2}(L_{0}-L)u)\in
W^{2,2}_{1-\frac{n}{2}+\alpha}\subset W^{2,2}_{\mu},$

for any $\mu<\mu^{\prime}\leq 1-\frac{n}{2}+\alpha$. Consequently, (8.62)
implies that $u\in W^{2,2}_{\mu},$ for any $\mu<\mu^{\prime}$, and Claim
$(\ref{eqn: W-functional claim})$ follows.

If $\mu^{\prime}>1-\frac{n}{2}+\alpha$, then $u\in
W^{2,2}_{1-\frac{n}{2}+\alpha}$. Starting from this regularity estimate for
$u$, and applying the above process, we obtain

(8.75) $Q(r^{2}u\ln u)+Q(r^{2}(L_{0}-L)u)\in W^{2,2}_{1-\frac{n}{2}+2\alpha}.$

Now if $\mu^{\prime}\leq 1-\frac{n}{2}+2\alpha$, then we obtain $u\in
W^{2,2}_{\mu}$, for any $\mu<\mu^{\prime}$. Otherwise, we repeat the above
argument finitely many times to prove Claim $(\ref{eqn: W-functional claim})$.

Starting from the regularity estimate in Claim $(\ref{eqn: W-functional
claim})$, a weighted $L^{p}$ elliptic bootstrapping argument as in the proof
of Theorem 5.2 in [DW20] gives

(8.76) $|\nabla^{i}u|=o(r^{\mu-i})$

as $r\rightarrow 0$, for $i=0,1$ and any $\mu<\mu^{\prime}$.

Step 2. Finally, we estimate the exponent $\mu^{\prime}$ in the leading order
term of $R_{1}u$ to complete the proof. For that, we apply $r^{2}L_{0}$ to
$R_{1}u$ to obtain, via $(\ref{eqn: asymptotic R1u w-functional})$,

(8.77) $r^{2}L_{0}\circ
R_{1}u\sim\\{[-4\mu^{\prime}(\mu^{\prime}+n-2)-(n-1)(n-2)]a(y)+(-4\Delta_{\hat{g}}+R_{\hat{g}})a(y)\\}r^{\mu^{\prime}}(\ln
r)^{p},$

as $r\rightarrow 0$.

On the other hand, applying $r^{2}L_{0}$ to the equation (8.62), using the
equation $Lu=u\ln u$ and (8.25), one has

(8.78) $r^{2}L_{0}\circ R_{1}u=R_{2}(r^{2}u\ln u)+R_{2}(r^{2}(L_{0}-L)u)\in
W^{\infty,2}_{1-\frac{n}{2}+\alpha}.$

Now we separate the discussion into two cases:

Case 1: If

(8.79)
$[-4\mu^{\prime}(\mu^{\prime}+n-2)-(n-1)(n-2)]a(y)+(-4\Delta_{\hat{g}}+R_{\hat{g}})a(y)\equiv
0,$

then $a(y)$ is an eigenfunction of $-4\Delta_{\hat{g}}+R_{\hat{g}}$ and

(8.80) $\mu^{\prime}=-\frac{n-2}{2}+\frac{\sqrt{\nu^{\prime}-(n-2)}}{2},$

where $\nu^{\prime}$ is an eigenvalue of $-4\Delta_{\hat{g}}+R_{\hat{g}}$ on
$(F,\hat{g})$. And we are done. Here we have used Theorem 5.2 in [DW20],
together with $\nu^{\prime}>n-2$ via our assumption on the scalar curvature,
to rule out the other root.

Case 2: If

(8.81)