Convergence to the Product of the Standard Spheres and Eigenvalues of the
Laplacian

Convergence to the Product of the Standard Spheres
and Eigenvalues of the Laplacian††This paper is a contribution to the Special
Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th
Birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Gromov.html

Masayuki AINO

M. Aino

RIKEN, Center for Advanced Intelligence Project AIP,
1-4-1 Nihonbashi, Tokyo 103-0027, Japan<EMAIL_ADDRESS>https://sites.google.com/site/masayukiaino/

Received July 17, 2020, in final form February 07, 2021; Published online
February 24, 2021

We show a Gromov–Hausdorff approximation to the product of the standard
spheres $S^{n-p}\times S^{p}$ for Riemannian manifolds with positive Ricci
curvature under some pinching condition on the eigenvalues of the Laplacian
acting on functions and forms.

Gromov–Hausdorff distance; Lichnerowicz–Obata estimate; parallel $p$-form

53C20; 58J50

##  1 Introduction

In this article we show that if an $n$-dimensional closed Riemannian manifold
with positive Ricci curvature admits an almost parallel $p$-form ($2\leq
p<n/2$) in $L^{2}$-sense and if the first $n+1$ eigenvalues of the Laplacian
acting on functions are close to their optimal values, then the Riemannian
manifold is close to the product of the standard spheres $S^{n-p}\times S^{p}$
with appropriate radii (Main Theorem below). Before giving the precise
statement, we provide some backgrounds.

The Lichnerowicz–Obata theorem is one of the classical theorem about the first
eigenvalue of the Laplacian. Lichnerowicz showed the optimal comparison result
for the first eigenvalue when the Riemannian manifold has positive Ricci
curvature, and Obata showed that the equality of the Lichnerowicz estimate
implies that the Riemannian manifold is isometric to the standard sphere. In
the following, $\lambda_{k}(g)$ denotes the $k$-th positive eigenvalue of the
minus Laplacian $-\Delta:=-\operatorname{tr}_{g}\operatorname{Hess}$ acting on
functions.

######  Theorem 1.1 (Lichnerowicz–Obata theorem).

Take an integer $n\geq 2$. Let $(M,g)$ be an $n$-dimensional closed Riemannian
manifold. If $\operatorname{Ric}\geq(n-1)g$, then $\lambda_{1}(g)\geq n$. The
equality $\lambda_{1}(g)=n$ holds if and only if $(M,g)$ is isometric to the
standard sphere of radius $1$.

Petersen [26], Aubry [8] and Honda [21] showed the stability result of the
Lichnerowicz–Obata theorem. In the following, $d_{{\rm GH}}$ denotes the
Gromov–Hausdorff distance and $S^{n}$ denotes the $n$-dimensional standard
sphere of radius 1 (see Definition 2.13 for the definition of the
Gromov–Hausdorff distance).

######  Theorem 1.2 ([8, 21, 26]).

For given an integer $n\geq 2$ and a positive real number $\epsilon>0$, there
exists $\delta(n,\epsilon)>0$ such that if $(M,g)$ is an $n$-dimensional
closed Riemannian manifold with $\operatorname{Ric}\geq(n-1)g$ and
$\lambda_{n}(g)\leq n+\delta$, then $d_{{\rm
GH}}\big{(}M,S^{n}(1)\big{)}\leq\epsilon$.

Note that Petersen considered the pinching condition on $\lambda_{n+1}(g)$,
and Aubry and Honda improved it independently.

When the Riemannian manifold admits a non-trivial parallel differential form,
we have the stronger estimate.

######  Theorem 1.3 ([1, 18]).

Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. Assume that
$\operatorname{Ric}\geq(n-p-1)g$ and that there exists a nontrivial parallel
$p$-form on $M$ $(2\leq p\leq n/2)$. Then, we have $\lambda_{1}(g)\geq n-p.$
Moreover, if $p<n/2$ and $\lambda_{n-p+1}(g)=n-p$ hold, then $(M,g)$ is
isometric to a product $S^{n-p}(1)\times(X,g^{\prime})$, where
$(X,g^{\prime})$ is some $p$-dimensional closed Riemannian manifold.

To simplify the numbers appearing in the theorem, we consider the assumption
$\operatorname{Ric}\geq$ $(n-p-1)g$ instead of $\operatorname{Ric}\geq(n-1)g$.
By scaling, the estimate in Theorem 1.1 becomes $\lambda_{1}(g)\geq
n(n-p-1)/(n-1)$ when $\operatorname{Ric}\geq(n-p-1)g$. Note that we have
$n-p>n(n-p-1)/(n-1)$.

To state the almost version of Theorem 1.3, we introduce the first eigenvalue
of the connection Laplacian acting on $p$-forms $\lambda_{1}(\Delta_{C,p})$
for a closed Riemannian manifold $(M,g)$:

$\lambda_{1}(\Delta_{C,p}):=\inf\left\\{\frac{\|\nabla\omega\|_{L^{2}}^{2}}{\|\omega\|_{L^{2}}^{2}}\colon\omega\in\Gamma\Big{(}\bigwedge^{p}T^{\ast}M\Big{)}\text{
with }\omega\neq 0\right\\}.$

Note that there exists a non zero $p$-form $\omega$ with
$\|\nabla\omega\|_{L^{2}}^{2}\leq\delta\|\omega\|_{L^{2}}^{2}$ for some
$\delta>0$ if and only if $\lambda_{1}(\Delta_{C,p})\leq\delta$ holds. For
arbitrary integers $n$, $p$ with $2\leq p\leq n/2$ and a real number
$\epsilon>0$, considering a small perturbation of $S^{n-p}(1)\times
S^{p}(r_{n,p})$, we can find an $n$-dimensional closed Riemannian manifold
with $\operatorname{Ric}\geq(n-p-1)g$ such that
$0<\lambda_{1}(\Delta_{C,p})<\epsilon$ holds. Here we defined
$r_{n,p}:=\sqrt{(p-1)/(n-p-1)}$. In other words, we do not have the gap
theorem for the first eigenvalue of the connection Laplacian
$\lambda_{1}(\Delta_{C,p})$ if we only assume a lower Ricci curvature bound.

Let us state the almost version of the eigenvalue estimate.

######  Theorem 1.4 ([1]).

For given integers $n\geq 4$ and $2\leq p\leq n/2$, there exists a constant
$C(n,p)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian
manifold with $\operatorname{Ric}_{g}\geq(n-p-1)g$, then we have

$\displaystyle\lambda_{1}(g)\geq n-p-C(n,p)\lambda_{1}(\Delta_{C,p})^{1/2}.$

This theorem recovers the estimate in Theorem 1.3 when
$\lambda_{1}(\Delta_{C,p})=0$.

We next state the approximation result to the product space.

######  Theorem 1.5 ([1]).

For given integers $n\geq 5$ and $2\leq p<n/2$ and a positive real number
$\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that if
$(M,g)$ is an $n$-dimensional closed Riemannian manifold with
$\operatorname{Ric}_{g}\geq(n-p-1)g$,

$\displaystyle\lambda_{n-p+1}(g)\leq n-p+\delta$

and

$\displaystyle\lambda_{1}(\Delta_{C,p})\leq\delta,$

then $M$ is orientable and

$d_{{\rm GH}}\big{(}M,S^{n-p}(1)\times X\big{)}\leq\epsilon,$

where $X$ is some compact metric space.

In this article we study the structure of the metric space $X$ in this theorem
and show that $X$ with some appropriate Borel measure satisfies the ${\rm
RCD}^{\ast}(n-p-1,p)$ condition (see Proposition 3.2), which means a synthetic
notion of “$\operatorname{Ric}\geq n-p-1$ and $\dim\leq p$ with Riemannian
structure” (see Definition 2.6). As a consequence, we can show the estimate
$\lambda_{n-p+2}(g)\geq p(n-p-1)/(p-1)-\epsilon$ under the assumption of
Theorem 1.5 (see Theorem 4.1) and the following theorem.

######  Main Theorem.

For given integers $n\geq 5$ and $2\leq p<n/2$ and a positive real number
$\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that if
$(M,g)$ is an $n$-dimensional closed Riemannian manifold with
$\operatorname{Ric}_{g}\geq(n-p-1)g$,

$\displaystyle\lambda_{n-p+1}(g)\leq
n-p+\delta,\qquad\lambda_{n+1}(g)\leq\frac{p(n-p-1)}{p-1}+\delta$

and

$\displaystyle\lambda_{1}(\Delta_{C,p})\leq\delta,$

then

$d_{{\rm GH}}\bigg{(}M,S^{n-p}(1)\times
S^{p}\bigg{(}\sqrt{\frac{p-1}{n-p-1}}\bigg{)}\bigg{)}\leq\epsilon,$

and $M$ is diffeomorphic to $S^{n-p}\times S^{p}$.

We have the last assertion by the topological stability theorem due to
Cheeger–Colding [12, Theorem A.1.12]. We show the main theorem including the
case when $\lambda_{1}(\Delta_{C,n-p})\leq\delta$ (see Theorem 4.2).

##  2 Preliminaries

###  2.1 Basic notation

We first recall some basic definitions and fix our convention.

Let $(M,g)$ be a closed Riemannian manifold. For any $p\geq 1$, we use the
normalized $L^{p}$-norm:

$\displaystyle\|f\|_{L^{p}}^{p}:=\frac{1}{\operatorname{Vol}(M)}\int_{M}|f|^{p}\,{\rm
d}\mu_{g},$

and $\|f\|_{L^{\infty}}:=\mathop{\mathrm{ess~{}sup}}\limits_{x\in M}|f(x)|$
for a measurable function $f$ on $M$. We also use these notation for tensors.
We have $\|f\|_{L^{p}}\leq\|f\|_{L^{q}}$ for any $p\leq q\leq\infty$.

Let $\nabla$ denote the Levi-Civita connection. Throughout this paper,
$0=\lambda_{0}(g)<\lambda_{1}(g)\leq\lambda_{2}(g)\leq\cdots\to\infty$ denotes
the eigenvalues of the minus Laplacian
$-\Delta=-\operatorname{tr}\operatorname{Hess}$ acting on functions counted
with multiplicities. For $p=0,1,\ldots,n$, let

$\lambda_{1}(\Delta_{C,p}):=\inf\left\\{\frac{\|\nabla\omega\|_{L^{2}}^{2}}{\|\omega\|_{L^{2}}^{2}}\colon\omega\in\Gamma\Big{(}\bigwedge^{p}T^{\ast}M\Big{)}\text{
with }\omega\neq 0\right\\}.$

For metric space $(X,d)$ and $k\in\mathbb{R}_{\geq 0}$, let $\mathcal{H}^{k}$
denote the $k$-dimensional Hausdorff measure. If
$0<\mathcal{H}^{k}(X)<\infty$, let $\underline{\mathcal{H}}^{k}$ denote the
normalized $k$-dimensional Hausdorff measure:

$\underline{\mathcal{H}}^{k}:=\frac{1}{\mathcal{H}^{k}(X)}\mathcal{H}^{k}.$

In this article, for metric spaces $(X_{i},d_{i})$ ($i=1,2$), let $d_{1}\times
d_{2}$ denote the distance on $X_{1}\times X_{2}$ satisfying

$(d_{1}\times
d_{2})^{2}\left((x_{1},x_{2}),(y_{1},y_{2})\right)=d_{1}^{2}(x_{1},y_{1})+d_{2}^{2}(x_{2},y_{2})$

for $(x_{1},x_{2}),(y_{1},y_{2})\in X_{1}\times X_{2}$.

###  2.2 Metric measure spaces

In this article we only consider a compact metric measure space with full
support and unit total mass for simplicity of the description because it is
enough for our purpose.

######  Definition 2.1.

In this article we say that $(X,d,\mathfrak{m})$ is a compact metric measure
space if $(X,d)$ is a compact metric space and $\mathfrak{m}$ is a Borel
measure with $\operatorname{supp}\mathfrak{m}=X$ and $\mathfrak{m}(X)=1$.

We introduce some functional analytic tools on a metric measure space. Our
main references are [2, 15, 16].

######  Definition 2.2.

Let $(X,d,\mathfrak{m})$ be a compact metric measure space.

  * •

_Local Lipschitz constant_. Let $\operatorname{LIP}(X)$ denote the set of the
Lipschitz functions on $X$. For each $f\in\operatorname{LIP}(X)$ and $x\in X$,
we define a local Lipschitz constant $\operatorname{Lip}(f)(x)$ by

$\operatorname{Lip}(f)(x):=\limsup_{y\to x}\frac{|f(x)-f(y)|}{d(x,y)}$

if $x\in X$ is not an isolated point, and $\operatorname{Lip}(f)(x)=0$
otherwise.

  * •

_Cheeger energy_. For each $f\in L^{2}(X)$, we define the Cheeger energy
$\operatorname{Ch}(f)\in[0,\infty]$ by

$\operatorname{Ch}(f):=\frac{1}{2}\inf\left\\{\liminf_{i\to\infty}\int_{X}(\operatorname{Lip}(f_{i}))^{2}\,{\rm
d}\mathfrak{m}\colon f_{i}\in\operatorname{LIP}(X)\text{ and
}\lim_{i\to\infty}\|f-f_{i}\|_{L^{2}}=0\right\\}.$

Define

$W^{1,2}(X)=W^{1,2}(X,d,\mathfrak{m}):=\\{f\in
L^{2}(X)\colon\operatorname{Ch}(f)<\infty\\}.$

We have that $W^{1,2}(X)$ is a Banach space with the norm
$\|f\|_{W^{1,2}}=(\|f\|_{L^{2}}^{2}+2\operatorname{Ch}(f))^{1/2}$.

  * •

_Minimal relaxed gradient_. We say that $|Df|\in L^{2}(X)$ is the minimal
relaxed gradient of $f\in W^{1,2}(X)$ if there exists a sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ of Lipschitz function such that
$\lim\limits_{i\to\infty}\|f-f_{i}\|_{L^{2}}=0$,
$\lim\limits_{i\to\infty}\|\,|Df|-\operatorname{Lip}(f_{i})\|_{L^{2}}=0$ and

$\operatorname{Ch}(f)=\frac{1}{2}\int_{X}|Df|^{2}\,{\rm d}\mathfrak{m}.$

For any $f\in W^{1,2}(X)$, the minimal relaxed gradient $|Df|\in L^{2}(X)$
exists and unique. See [2, Definition 4.2 and Lemma 4.3].

  * •

_Sobolev-to-Lipschitz property._ We say that $(X,d,\mathfrak{m})$ satisfies
the Sobolev-to-Lipschitz property if any $f\in W^{1,2}(X)$ with $|Df|\leq 1$
$\mathfrak{m}$-a.e. in $X$ is a $1$-Lipschitz function on $X$ (more precisely,
$f$ has a $1$-Lipschitz representative).

  * •

_Infinitesimally Hilbertian_. We say that $(X,d,\mathfrak{m})$ is
infinitesimally Hilbertian if $\operatorname{Ch}$ is a quadratic form. This
condition holds if and only if $\big{(}W^{1,2}(X),\|\cdot\|_{W^{1,2}}\big{)}$
is a Hilbert space. In this case, we define $\mathcal{E}\colon
W^{1,2}(X)\times W^{1,2}(X)\to\mathbb{R}$ by

$\mathcal{E}(f,g)=\frac{1}{2}(\operatorname{Ch}(f+g)-\operatorname{Ch}(f-g)).$

  * •

_Laplacian_. If $(X,d,\mathfrak{m})$ is infinitesimally Hilbertian, then we
define

$\mathcal{D}(\Delta):=\left\\{f\in
W^{1,2}(X)\colon\begin{array}[]{@{}l@{}}\text{there exists $\Delta f\in
L^{2}(X)$ such that we have}\\\ \text{$\mathcal{E}(f,g)=-\int_{X}g\Delta
f\,{\rm d}\mathfrak{m}$ for any $g\in W^{1,2}(X)$}\end{array}\right\\}.$

For each $f\in\mathcal{D}(\Delta)$, $\Delta f\in L^{2}(X)$ is uniquely
determined.

  * •

_The function_ $\langle Df_{1},Df_{2}\rangle$. If $(X,d,\mathfrak{m})$ is
infinitesimally Hilbertian, then we define $\langle Df_{1},Df_{2}\rangle\in
L^{1}(X)$ for $f_{1},f_{2}\in W^{1,2}(X)$ by

$\langle Df_{1},Df_{2}\rangle=\lim_{\epsilon\to
0}\frac{1}{2\epsilon}(|D(f_{1}+\epsilon f_{2})|^{2}-|Df_{1}|^{2})\in
L^{1}(X).$

This notion is well-defined by the convexity of the minimal relaxed gradient
(see [3, Definition 4.12]). We have that $\langle
Df_{1},Df_{1}\rangle=|Df_{1}|^{2}$ $\mathfrak{m}$-a.e. in $X$, that $|\langle
Df_{1},Df_{2}\rangle|\leq|Df_{1}||Df_{2}|$ $\mathfrak{m}$-a.e. in $X$, that
$\langle\cdot,\cdot\rangle$ is a symmetric bilinear form, and that

$\mathcal{E}(f_{1},f_{2})=\int_{X}\langle Df_{1},Df_{2}\rangle\,{\rm
d}\mathfrak{m}$

by [3, Propositions 4.13 and 4.14, Theorem 4.18].

  * •

_Heat flow_. Let $(X,d,\mathfrak{m})$ be infinitesimally Hilbertian. Let
$\\{P_{t}f\\}_{t>0}$ denote the gradient flow of the Cheeger energy
$\operatorname{Ch}$ starting from $f\in L^{2}(X)$ (see [16, Definition
5.2.5]). The flow $\\{P_{t}f\\}_{t>0}$ is called the heat flow and
characterized as the unique $C^{1}$ map $(0,\infty)\to L^{2}(X)$ (it turns out
to be $C^{\infty}$ [16, Proposition 5.2.12]) satisfying the following
conditions (see [16, Theorem 5.1.12]):

    * –

We have $P_{t}f\to f$ strongly in $L^{2}(X)$ as $t\to 0$.

    * –

For each $t>0$, we have that $P_{t}f\in\mathcal{D}(\Delta)$ and that

$\frac{\rm d}{{\rm d}t}P_{t}f=\Delta P_{t}f$

in $L^{2}(X)$.

Moreover, we have the following properties (see [16, Section 5.2.2]):

    * –

For each $t>0$ and $f\in L^{2}(X)$, we have

$\displaystyle\operatorname{Ch}(P_{t}f)\leq\inf_{g\in
W^{1,2}(X)}\left(\operatorname{Ch}(g)+\frac{\|f-g\|_{L^{2}}^{2}}{2t}\right),$
$\displaystyle\|\Delta
P_{t}f\|_{L^{2}}^{2}\leq\inf_{g\in\mathcal{D}(\Delta)}\left(\|\Delta
g\|_{L^{2}}^{2}+\frac{\|f-g\|_{L^{2}}^{2}}{t^{2}}\right).$

See also [16, Theorem 5.1.12].

    * –

For each $t>0$, $P_{t}\colon L^{2}(X)\to L^{2}(X)$ is a linear map satisfying

$\int_{X}gP_{t}f\,{\rm d}\mathfrak{m}=\int_{X}fP_{t}g\,{\rm d}\mathfrak{m}$

for any $f,g\in L^{2}(X)$.

    * –

For each $s,t>0$, we have $P_{s+t}=P_{s}\circ P_{t}$.

    * –

For each $f\in\mathcal{D}(\Delta)$ and $s>0$, we have that

$\lim_{t\to 0+}\frac{P_{t}f-f}{t}=\Delta f$

in $L^{2}(X)$ and that $\Delta P_{s}f=P_{s}\Delta f$.

    * –

For each $t>0$, $c\in\mathbb{R}$ and $f\in L^{2}(X)$ with $f\leq c$
$\mathfrak{m}$-a.e. in $X$, we have $P_{t}f\leq c$ $\mathfrak{m}$-a.e. in $X$.

    * –

For each $t>0$, $p\in[1,\infty)$ and $f\in L^{2}(X)\cap L^{p}(X)$, we have
$\|P_{t}f\|_{L^{p}}\leq\|f\|_{L^{p}}$. In particular, we can extend the map
$P_{t}\colon L^{2}(X)\cap L^{p}(X)\to L^{2}(X)\cap L^{p}(X)$ to $P_{t}$:
$L^{p}(X)\to L^{p}(X)$.

We can also show the following properties by the above properties:

    * –

For each $f\in W^{1,2}(X)$, we have $\operatorname{Ch}(P_{t}f-f)\to 0$ as
$t\to 0$. Indeed, the properties
$\operatorname{Ch}(P_{t}f)\leq\operatorname{Ch}(f)$ and $P_{t}f\to f$ in
$L^{2}$ as $t\to 0$ imply that $P_{t}f$ converges to $f$ weakly in $W^{1,2}$,
and so $\limsup\limits_{t\to
0}\operatorname{Ch}(P_{t}f)\leq\operatorname{Ch}(f)$ implies that $P_{t}f$
converges to $f$ strongly in $W^{1,2}$ as $t\to 0$.

    * –

For each $p\in[1,\infty)$ and $f\in L^{p}(X)$, we have
$\|P_{t}f-f\|_{L^{p}}\to 0$ as $t\to 0$. This can be verified by applying the
above properties to the truncated function $f_{1}=\max\\{\min\\{f,c\\},-c\\}$
for a sufficiently large constant $c\\!>\\!0$ and its remaining part
$f\\!-\\!f_{1}$.

  * •

_Test functions_. Let $(X,d,\mathfrak{m})$ be infinitesimally Hilbertian. We
define

$\operatorname{TestF}(X):=\big{\\{}f\in\mathcal{D}(\Delta)\cap
L^{\infty}(X)\colon|\nabla f|\in L^{\infty}(X)\text{ and }\Delta f\in
W^{1,2}(X)\big{\\}}.$

  * •

_Pre-cotangent module_. We define

$\operatorname{PCM}:=\left\\{\\{(f_{i},A_{i})\\}\colon\,\begin{array}[]{@{}l@{}}\text{$\\{A_{i}\\}_{i=1}^{\infty}$
is a pairwise disjoint family of Borel subsets of $X$}\\\ \text{with
$\bigcup_{i}A_{i}=X$, $f_{i}\in W^{1,2}(X)$ with
$\sum_{i}\int_{A_{i}}|Df_{i}|^{2}\,{\rm
d}\mathfrak{m}<\infty$}\end{array}\right\\}.$

We say that $\\{(f_{i},A_{i})\\}\in\operatorname{PCM}$ is equivalent to
$\\{(g_{i},B_{i})\\}\in\operatorname{PCM}$ (denote it by
$\\{(f_{i},A_{i})\\}\sim\\{(g_{i},B_{i})\\}$) if

$|D(f_{i}-g_{j})|=0\text{ $\mathfrak{m}$-a.e.\ in $A_{i}\cap B_{j}$ for each
$i,j\in\mathbb{Z}_{>0}$}.$

We define

$\displaystyle|\cdot|\colon\operatorname{PCM}/{\sim}\to
L^{2}(X),\qquad[\\{(f_{i},A_{i})\\}]\mapsto\sum_{i=1}^{\infty}\chi_{A_{i}}|Df_{i}|,$
$\displaystyle\|\cdot\|_{L^{2}}\colon\operatorname{PCM}/{\sim}\to[0,\infty),\quad[\\{(f_{i},A_{i})\\}]\mapsto\big{\|}|[\\{(f_{i},A_{i})\\}]|\big{\|}_{L^{2}}\\!=\\!\bigg{(}\\!\sum_{i}\\!\int_{A_{i}}\\!\\!|Df_{i}|^{2}\,{\rm
d}\mathfrak{m}\\!\bigg{)}^{1/2}\\!\\!\\!,$

where $\chi_{A_{i}}$ denotes the characteristic function. Then,
$(\operatorname{PCM}/{\sim},\|\cdot\|_{L^{2}})$ is naturally equipped with the
structure of the normed vector space. Moreover, we define

$\bigg{(}\sum_{i=1}^{\infty}a_{i}\chi_{A_{i}}\bigg{)}\cdot[\\{(f_{i},B_{i})\\}]:=[\\{(a_{i}f_{j},A_{i}\cap
B_{j})\\}_{i,j}]$

for each $\sum_{i=1}^{\infty}a_{i}\chi_{A_{i}}\in\operatorname{Sf}(X)$ and
$[\\{(f_{i},B_{i})\\}]\in\operatorname{PCM}/{\sim}$, where
$\operatorname{Sf}(X)$ is defined by

$\operatorname{Sf}(X):=\left\\{\sum_{i=1}^{\infty}a_{i}\chi_{A_{i}}\colon\begin{array}[]{@{}l@{}}\text{$\\{A_{i}\\}_{i=1}^{\infty}$
is a pairwise disjoint family of Borel subsets of $X$}\\\ \text{with
$\bigcup_{i}A_{i}=X$, $a_{i}\in\mathbb{R}$ with
$\sup_{i}|a_{i}|<\infty$}\end{array}\right\\}.$

Then, we have
$\|f\cdot\omega\|_{L^{2}}\leq\|f\|_{L^{\infty}}\|\omega\|_{L^{2}}$ for each
$f\in\operatorname{Sf}(X)$ and $\omega\in\operatorname{PCM}/{\sim}$.

  * •

_Cotangent module_. We define the cotangent module $L^{2}(T^{\ast}X)$ as a
completion of the normed vector space
$(\operatorname{PCM}/{\sim},\|\cdot\|_{L^{2}})$. We can extend the action
$\operatorname{Sf}(X)\times\operatorname{PCM}/{\sim}\to\operatorname{PCM}/{\sim}$
to $L^{\infty}(X)\times L^{2}(T^{\ast}X)\to L^{2}(T^{\ast}X)$, and
$|\cdot|\colon\operatorname{PCM}/{\sim}\to L^{2}(X)$ to $|\cdot|\colon
L^{2}(T^{\ast}X)\to L^{2}(X)$. Then, $L^{2}(T^{\ast}X)$ is equipped with the
structure of an $L^{2}$-normed $L^{\infty}(X)$ module, i.e., we have that

$\displaystyle(fg)\cdot\omega=f\cdot(g\cdot\omega),\qquad
1\cdot\omega=\omega,\qquad|\omega|\geq 0\quad\text{$\mathfrak{m}$-a.e.\ in
X},$
$\displaystyle\|\omega\|_{L^{2}}=\big{\|}|\omega|\big{\|}_{L^{2}},\qquad|f\cdot\omega|=|f||\omega|$

for each $f,g\in L^{\infty}(X)$ and $\omega\in L^{2}(T^{\ast}X)$. Note that we
use $W^{1,2}(X)$ instead of the Sobolev class $S^{2}(X)$ (see [15, Definition
2.1.4]) in the definition of $\operatorname{PCM}$. However, we can approximate
elements of $S^{2}(X)$ by elements of $W^{1,2}(X)$ (see [15, Proposition
2.2.5]), and so our definition of $L^{2}(T^{\ast}X)$ coincides with [15,
Definition 2.2.1]. In general, $L^{2}(T^{\ast}X)$ does not need to be a
Hilbert space, and $L^{2}(T^{\ast}X)$ is a Hilbert space if and only if
$(X,d,\mathfrak{m})$ is infinitesimally Hilbertian [15, Proposition 2.3.17].

  * •

_Pointwise scalar product_. If $(X,d,\mathfrak{m})$ is infinitesimally
Hilbertian, we can define a pointwise scalar product

$\langle[\\{(f_{i},A_{i})\\}],[\\{(g_{i},B_{i})\\}]\rangle:=\sum_{i,j=1}^{\infty}\chi_{A_{i}\cap
B_{j}}\langle Df_{i},Dg_{j}\rangle$

for each
$[\\{(f_{i},A_{i})\\}],[\\{(g_{i},B_{i})\\}]\in\operatorname{PCM}/{\sim}$ and
extend it to $\langle\cdot,\cdot\rangle\colon L^{2}(T^{\ast}X)\times
L^{2}(T^{\ast}X)\to L^{1}(X)$. Then, $\langle\cdot,\cdot\rangle$ is symmetric
and $L^{2}(T^{\ast}X)$ is a Hilbert space with the inner product defined by

$(\omega,\eta)\mapsto\int_{X}\langle\omega,\eta\rangle\,{\rm d}\mathfrak{m}$

for each $\omega,\eta\in L^{2}(T^{\ast}X)$. Clearly, for each $\omega,\eta\in
L^{2}(T^{\ast}X)$, we have $\langle\omega,\omega\rangle=|\omega|^{2}$ and
$|\langle\omega,\eta\rangle|\leq|\omega||\eta|$ $\mathfrak{m}$-a.e. in $X$.

  * •

_Differential_. We define the differential ${\rm d}\colon W^{1,2}(X)\to
L^{2}(T^{\ast}X)$ by ${\rm d}f=[(f,X)]$ for each $f\in W^{1,2}(X)$. Clearly,
we have that $|Df|=|{\rm d}f|\in L^{2}(X)$ for each $f\in W^{1,2}(X)$.
Moreover, if $(X,d,\mathfrak{m})$ is infinitesimally Hilbertian, then we have
$\langle Df,Dg\rangle=\langle{\rm d}f,{\rm d}g\rangle\in L^{1}(X)$ for each
$f,g\in W^{1,2}(X)$.

  * •

_Tangent module_. We define the tangent module $L^{2}(TX)$ by

$L^{2}(TX):=\left\\{V\colon\begin{array}[]{@{}l@{}}\text{$V\colon
L^{2}(T^{\ast}X)\to L^{1}(X)$ is a bounded linear operator such that}\\\
\text{$V(f\cdot\omega)=fV(\omega)$ holds for all $f\in L^{\infty}(X)$ and
$\omega\in L^{2}(T^{\ast}X)$}\end{array}\right\\}.$

If $(X,d,\mathfrak{m})$ is infinitesimally Hilbertian, then the map

$L^{2}(T^{\ast}X)\to L^{2}(TX),\qquad\omega\mapsto\langle\omega,\cdot\rangle$

is bijective (see [15, Theorem 1.2.24]). Under this identification,
$L^{2}(TX)$ is equipped with the structure of an $L^{2}$-normed
$L^{\infty}(M)$ module and a pointwise scalar product
$\langle\cdot,\cdot\rangle\colon L^{2}(TX)\times L^{2}(TX)\to L^{1}(X)$. Note
that even if $(X,d,\mathfrak{m})$ is not infinitesimally Hilbertian, the
tangent module $L^{2}(TX)$ is naturally equipped with the structure of an
$L^{2}$-normed $L^{\infty}(M)$ module (see [15, Definition 1.2.6 and
Proposition 1.2.14]).

  * •

_Gradient_. If $(X,d,\mathfrak{m})$ is infinitesimally Hilbertian, we define a
map $\nabla\colon W^{1,2}(X)\to L^{2}(TX)$ by $\nabla f=\langle{\rm
d}f,\cdot\rangle$ for each $f\in W^{1,2}(X)$. Clearly, we have that $|\nabla
f|=|Df|=|{\rm d}f|\in L^{2}(X)$ and $\langle\nabla f,\nabla g\rangle=\langle
Df,Dg\rangle\in L^{1}(X)$ for each $f,g\in W^{1,2}(X)$.

  * •

_Divergence_. If $(X,d,\mathfrak{m})$ is infinitesimally Hilbertian, then we
define

$\mathcal{D}(\operatorname{div}):=\left\\{V\in
L^{2}(TX)\colon\begin{array}[]{@{}l@{}}\text{there exists
$\operatorname{div}V\in L^{2}(X)$ such that we have}\\\ \text{$\int_{X}\langle
V,\nabla g\rangle\,{\rm d}\mathfrak{m}=-\int_{X}g\,\operatorname{div}V\,{\rm
d}\mathfrak{m}$ for any $g\in W^{1,2}(X)$}\end{array}\right\\}.$

For each $V\in\mathcal{D}(\operatorname{div})$, $\operatorname{div}V\in
L^{2}(X)$ is uniquely determined.

  * •

_Symmetric part of the covariant derivative_ [7, Definition 5.4]. Let
$(X,d,\mathfrak{m})$ be infinitesimally Hilbertian. For a vector field
$V\in\mathcal{D}(\operatorname{div})$ we write $D^{\operatorname{sym}}V\in
L^{2}(X)$ if there exists $c>0$ such that we have

$\displaystyle\left|\int_{X}\langle V,\nabla f\rangle\Delta g+\langle V,\nabla
g\rangle\Delta f-\langle\nabla f,\nabla g\rangle\operatorname{div}V\,{\rm
d}\mathfrak{m}\right|\leq c\|\nabla f\|_{L^{4}}\|\nabla g\|_{L^{4}}$ (2.1)

for any $f,g\in\mathcal{D}(\Delta)$ with $|\nabla f|,|\nabla g|\in L^{4}(X)$
and $\Delta f,\Delta g\in L^{4}(X)$.

###  2.3 The RCD∗ condition and some properties

In this subsection we recall the definition of the ${\rm RCD}^{\ast}(K,N)$
space and its properties.

######  Definition 2.3.

We say that an infinitesimally Hilbertian metric measure space
$(X,d,\mathfrak{m})$ satisfies the Bakry–Émery condition ${\rm BE}(K,N)$ with
$K\in\mathbb{R}$ and $N\in[1,\infty)$ if for all $u\in\mathcal{D}(\Delta)$
with $\Delta u\in W^{1,2}(X)$ and all $\phi\in\mathcal{D}(\Delta)\cap
L^{\infty}(X)$ with $\phi\geq 0$ and $\Delta\phi\in L^{\infty}(X)$, we have

$\displaystyle\frac{1}{2}\int_{X}\Delta\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}\geq\int_{X}\phi\bigg{(}\langle\nabla\Delta u,\nabla
u\rangle+K|\nabla u|^{2}+\frac{1}{N}(\Delta u)^{2}\bigg{)}\,{\rm
d}\mathfrak{m}.$

######  Definition 2.4.

We say that an infinitesimally Hilbertian metric measure space
$(X,d,\mathfrak{m})$ satisfies the Bakry–Ledoux condition ${\rm BL}(K,N)$ with
$K\in\mathbb{R}$ and $N\in[1,\infty)$ if for all $u\in W^{1,2}(X)$ and $t>0$
we have

$|\nabla P_{t}f|^{2}+\frac{2tC(t)}{N}|\Delta P_{t}f|^{2}\leq{\rm
e}^{-2Kt}P_{t}\big{(}|\nabla f|^{2}\big{)}$

$\mathfrak{m}$-a.e. in X, where $C(t)>0$ is a function satisfying
$C(t)=1+O(t)$ as $t\to 0$.

######  Theorem 2.5 ([14, Theorem 4.8 and Proposition 4.9]).

An infinitesimally Hilbertian metric measure space $(X,d,m)$ satisfies the
${\rm BE}(K,N)$ condition if and only if it satisfies the ${\rm BL}(K,N)$
condition.

Let us recall an equivalent version of the definition of the ${\rm
RCD}^{\ast}(K,N)$ condition (see [14, Theorem 7 and Definition 3.16] for the
equivalence). If the total mass is finite, the ${\rm RCD}^{\ast}(K,N)$
condition is equivalent to the $\mathrm{RCD}(K,N)$ condition by [10, Corollary
13.7].

######  Definition 2.6.

We say that a compact infinitesimally Hilbertian metric measure space
$(X,d,\mathfrak{m})$ satisfies the ${\rm RCD}^{\ast}(K,N)$ condition with
$K\in\mathbb{R}$ and $N\in[1,\infty)$ if $(X,d,\mathfrak{m})$ satisfies the
${\rm BE}(K,N)$ condition and the Sobolev-to-Lipschitz property.

For more general metric measure space, we add the volume growth assumption to
Definition 2.6. However, it is automatically satisfied in our situation
because we assume that $\mathfrak{m}(X)=1$. Note that Definition 2.6 implies
that $(X,d)$ is a geodesic space by [4, Theorems 3.9 and 3.10] and [9, Theorem
2.5.23]. The original definition also implies this property (see [14, Remark
3.8] and [31, Remark 4.6]).

The definition of the ${\rm RCD}^{\ast}(K,N)$ condition is consistent to the
smooth case.

######  Proposition 2.7 ([14, Proposition 4.21]).

For any $n$-dimensional closed Riemannian manifold $(M,g)$ and real numbers
$K\in\mathbb{R}$ and $N\in[1,\infty)$, we have that
$(M,d_{g},\underline{\mathcal{H}}^{n})$ satisfies the ${\rm RCD}^{\ast}(K,N)$
condition if and only if $\operatorname{Ric}\geq Kg$ and $n\leq N$ hold.

Let us compare the local Lipschitz constant and the minimal relaxed gradient
for Lipschitz functions. If the ${\rm RCD}^{\ast}(K,N)$ condition holds, then
we have the doubling condition [32, Corollary 2.4] and the weak Poincaré
inequality [28, Theorem 1.1]. Moreover, our minimal relaxed gradient coincides
with Cheeger’s minimal generalized upper gradient [11, Definition 2.9] by [2,
Theorem 6.2]. Thus, we have the following theorem by [11, Theorem 6.1]:

######  Theorem 2.8.

Let $(X,d,\mathfrak{m})$ be a compact metric measure space satisfying the
${\rm RCD}^{\ast}(K,N)$ condition. Then, for any $f\in\operatorname{LIP}(X)$,
we have $\operatorname{Lip}f=|\nabla f|$ $\mathfrak{m}$-a.e. in $X$.

Under the ${\rm RCD}^{\ast}(K,N)$ condition, we have that

$\operatorname{TestF}(X)=\left\\{f\in\mathcal{D}(\Delta)\cap\operatorname{LIP}(X)\colon\Delta
f\in W^{1,2}(X)\right\\}.$

Let us make a remark on the heat kernel. Let $(X,d,\mathfrak{m})$ be a compact
metric measure space satisfying the ${\rm RCD}^{\ast}(K,N)$ condition. Then,
$\mathcal{E}$ is a strongly local Dirichlet form on $(X,\mathfrak{m})$ by [2,
Proposition 4.8] and [3, Proposition 4.11], and we have

$d(x,y)=\sup\\{|f(x)-f(y)|\colon\text{$f\in\operatorname{LIP}(X)$ with
$|\nabla f|\leq 1$ $\mathfrak{m}$-a.e.\ in $X$}\\}$

by [4, Theorem 3.9]. Since we have the doubling condition by [32, Corollary
2.4] and the strong local $(2,2)$ Poincaré inequality [30, Property (Ic)] by
[28, Theorem 1.1] and [19, Theorem 1], we can apply [29, Proposition 2.3] and
[30, Proposition 3.1] (see also [30, Theorem 3.5]), and so there exists a
locally Hölder continuous function $p\colon(0,\infty)\times X\times
X\to\mathbb{R}$ such that

$P_{t}f(x)=\int_{X}p(t,x,y)f(y)\,{\rm d}\mathfrak{m}(y)$

holds for any $f\in L^{1}(X)$. By [23, Theorem 1.2], for any $\epsilon>0$,
there exist constants $C_{i}=C_{i}(\epsilon,K,N)>1$ such that

$\frac{C_{1}^{-1}}{\mathfrak{m}(B_{\sqrt{t}}(x))}\exp\bigg{(}\\!{-}\frac{d^{2}(x,y)}{(4-\epsilon)t}-C_{2}t\bigg{)}\leq
p(t,x,y)\leq\frac{C_{1}}{\mathfrak{m}(B_{\sqrt{t}}(x))}\exp\bigg{(}\\!{-}\frac{d^{2}(x,y)}{(4+\epsilon)t}+C_{2}t\bigg{)}$

holds for each $x,y\in X$ and $t>0$. Here, we defined $B_{r}(x):=\\{z\in
X\colon d(x,z)<r\\}$ for $x\in X$ and $r>0$. By this and the Bishop–Gromov
inequality [32, Theorem 2.3], we have the following:

  * •

For any $f\in L^{1}(X)$, we have

$\displaystyle\|P_{t}f\|_{L^{\infty}}\leq C(t)\|f\|_{L^{1}}.$ (2.2)

  * •

For any $f\in C(X)$, the function

$[0,\infty)\times X\to\mathbb{R},\qquad(t,x)\mapsto(P_{t}f)(x)$

is continuous. Here, we defined $P_{0}f:=f$.

For any $f\in W^{1,2}(X)$ and $t>0$, we have $P_{t}f\in\operatorname{LIP}(X)$
by (2.2) for $|\nabla f|^{2}\in L^{1}(X)$, the ${\rm BL}(K,N)$ condition and
the Sobolev-to-Lipschitz property, and so $P_{t}f\in\operatorname{TestF}(X)$
by $\Delta P_{t}f=P_{t/2}\Delta P_{t/2}f\in W^{1,2}(X)$. In particular,
$\operatorname{TestF}(X)\subset W^{1,2}(X)$ is dense. As a corollary, we have
$P_{t}f=P_{t/2}P_{t/2}f\in\operatorname{TestF}(X)$ for any $f\in L^{2}(X)$ and
$t>0$. Note that since we assumed that $(X,d,\mathfrak{m})$ is compact and
$\mathfrak{m}(X)=1$, we can skip the truncation procedure.

We next recall some basic facts about the spectrum of $-\Delta$ on a compact
metric measure space $(X,d,\mathfrak{m})$ satisfying the ${\rm
RCD}^{\ast}(K,N)$ condition. By [28, Theorem 1.1], [19, Theorem 1] and [20,
Theorem 8.1], the inclusion $W^{1,2}(X)\to L^{2}(X)$ is a compact operator.
Thus, the spectrum of $-\Delta$ is discrete and positive:

$0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots\to\infty$

as the smooth case. See also the proof of [14, Theorem 4.22]. Let
$\\{\phi_{i}\\}_{i=0}^{\infty}$ be the corresponding eigenfunctions. Then,

$\bigoplus_{i=0}^{\infty}\mathbb{R}\phi_{i}=\bigg{\\{}\sum_{i=0}^{k}a_{i}\phi_{i}\colon
k\in\mathbb{Z}_{\geq 0}\text{ and }a_{i}\in\mathbb{R}\
(i=0,\ldots,k)\bigg{\\}}$

is dense in $L^{2}(X)$.

Finally, let us recall the notion of the regular Lagrangian flow, which is a
flow for a vector field in a non-smooth setting. Although time dependent
vector fields are considered in [6], we only deal with time independent vector
fields because it is enough for our purpose.

######  Definition 2.9 ([6]).

Let $(X,d,\mathfrak{m})$ be a compact measure space satisfying the ${\rm
RCD}^{\ast}(K,N)$ condition and take $T>0$. We say that
$\operatorname{Fl}^{V}\colon[0,T]\times X\to X$ is a regular Lagrangian flow
for a vector field $V\in L^{2}(TX)$ if the following properties hold:

  * (i)

There exists a constant $C>0$ such that

$\big{(}\operatorname{Fl}^{V}_{s}\big{)}_{\ast}\mathfrak{m}\leq C\mathfrak{m}$

holds for any $s\in[0,T]$.

  * (ii)

For each $x\in X$, the curve

$[0,T]\to X,\qquad s\mapsto\operatorname{Fl}_{s}^{V}(x)$

is continuous and $\operatorname{Fl}_{0}^{V}=\operatorname{Id}_{X}$. Moreover,
the map $X\to C([0,T];X),\,x\mapsto\operatorname{Fl}^{V}(\cdot,x)$ is Borel
measurable. Here, $C([0,T];X)$ is equipped with the topology induced by the
uniformly convergence.

  * (iii)

For any Lipschitz function $f$ on $X$, we have that
$f\big{(}\operatorname{Fl}^{V}(\cdot,x)\big{)}\in W^{1,1}(0,T)$ for
$\mathfrak{m}$-a.e. $x\in X$ and

$\frac{\rm d}{{\rm d}t}f\big{(}(\operatorname{Fl}^{V}(t,x)\big{)}={\rm
d}f(V)\big{(}(\operatorname{Fl}^{V}(t,x)\big{)}$

for $\mathcal{L}^{1}\times\mathfrak{m}$-a.e. $(t,x)\in(0,T)\times X$.

Note that the Borel measurability in (ii) can be verified under the assumption
of the following existence theorem because we construct the flow using the
disintegration theorem in [6, Theorem 8.4] and [7, Theorem 7.8].

We use the following form of the result of [6]. See also [7].

######  Theorem 2.10 (Ambrosio–Trevisan [6]).

Let $(X,d,\mathfrak{m})$ be a metric measure space satisfying the ${\rm
RCD}^{\ast}(K,N)$ condition and take $T>0$. For any vector field
$V\in\mathcal{D}(\operatorname{div})$ with $D^{\operatorname{sym}}V\in
L^{2}(X)$ and $(\operatorname{div}V)^{-}\in L^{\infty}(X)$
$((\operatorname{div}V)^{-}$ denotes the negative part of the divergence
$\operatorname{div}V)$, a regular Lagrangian flow
$\operatorname{Fl}^{V}\colon[0,T]\times X\to X$ exists and unique, in the
sense that if $\widetilde{\operatorname{Fl}}^{V}$ is another flow, then for
$\mathfrak{m}$-a.e. $x\in X$ we have that
$\operatorname{Fl}^{V}(s,x)=\widetilde{\operatorname{Fl}}^{V}(s,x)$ for every
$s\in[0,T]$. Moreover, we have that

$\displaystyle\big{(}\operatorname{Fl}^{V}_{s}\big{)}_{\ast}\mathfrak{m}\leq\exp(T\|(\operatorname{div}V)^{-}\|_{L^{\infty}})\mathfrak{m}$
(2.3)

for all $s\in[0,T]$.

######  Remark 2.11.

Let us give some comments about which assertions in [6] correspond to Theorem
2.10. We set $\mathcal{A}:=\operatorname{LIP}(X)$. The concept of the regular
Lagrangian flow is closely related to the continuity equation

$\frac{\rm d}{{\rm d}t}u_{t}+\operatorname{div}(u_{t}V)=0.$

For $\overline{u}\in L^{\infty}(X)$ with $\overline{u}\geq 0$, there exists a
weakly continuous weak solution (in the duality with $\mathcal{A}$) $u\in
L^{\infty}([0,T];L^{\infty}(X))$ of the continuity equation with initial
condition $\overline{u}$ in the sense of [6, Definition 4.2] satisfying

$\|u_{t}\|_{L^{\infty}}\leq\|\overline{u}\|_{L^{\infty}}\exp(T\|(\operatorname{div}V)^{-}\|_{L^{\infty}})$

and $u_{t}\geq 0$ for each $t\in[0,T]$ by [6, Theorems 4.3 and 4.6] (see also
[7, Theorem 6.1]). Note that [6, Theorem 4.6] deals with approximated
solutions. However, since we get the solution of the continuity equation as a
weak* limit of them, we have the same estimate. Moreover, since we have the
$L^{4}-\Gamma$ inequality [6, Definition 5.1 and Corollary 6.3], the solution
is unique by [6, Theorem 5.4] (see also [7, Theorem 6.4]) putting $p=s=r=4$
and $q=2$. Thus, by [6, Theorem 9.2], we can apply [7, Theorem 7.7] (see also
[6, Theorem 8.3]), and so there exists a unique regular Lagrangian flow for
$V$. Moreover, its proof shows the estimate (2.3).

###  2.4 Gromov–Hausdorff convergence and functions

In this subsection we recall some properties about the Gromov–Hausdorff
convergence.

######  Definition 2.12 (Hausdorff distance).

Let $(X,d)$ be a metric space. For each point $x_{0}\in X$, subsets
$A,B\subset X$ and $r>0$, define

$\displaystyle d(x_{0},A):=\inf\\{d(x_{0},a)\colon a\in A\\},$ $\displaystyle
B_{r}(A):=\\{x\in X\colon d(x,A)<r\\},$ $\displaystyle d_{{\rm
H},d}(A,B):=\inf\\{\epsilon>0\colon A\subset B_{\epsilon}(B)\text{ and
}B\subset B_{\epsilon}(A)\\}.$

We call $d_{{\rm H},d}$ the Hausdorff distance.

The Hausdorff distance defines a metric on the collection of compact subsets
of $X$.

######  Definition 2.13 (Gromov–Hausdorff distance).

Let $(X,d_{X})$, $(Y,d_{Y})$ be metric spaces. Define

$d_{{\rm GH}}(X,Y):=\inf\left\\{d_{{\rm
H},d}(X,Y)\colon\begin{array}[]{@{}l@{}}\text{$d$ is a metric on $X\coprod Y$
such that}\\\ \text{$d|_{X}=d_{X}$ and $d|_{Y}=d_{Y}$}\end{array}\right\\}.$

The Gromov–Hausdorff distance defines a metric on the set of isometry classes
of compact metric spaces (see [27, Proposition 11.1.3]).

######  Definition 2.14 ($\epsilon$-Hausdorff approximation map).

Let $(X,d_{X})$, $(Y,d_{Y})$ be metric spaces. We say that a map $\psi\colon
X\to Y$ is an $\epsilon$-Hausdorff approximation map for $\epsilon>0$ if the
following two conditions hold.

  * (i)

For all $a,b\in X$, we have $|d_{X}(a,b)-d_{Y}(\psi(a),\psi(b))|<\epsilon$,

  * (ii)

$\psi(X)$ is $\epsilon$-dense in $Y$, i.e., for all $y\in Y$, there exists
$x\in X$ with $d_{Y}(\psi(x),y)<\epsilon$.

If there exists an $\epsilon$-Hausdorff approximation map from $X$ to $Y$,
then we can show that $d_{{\rm GH}}(X,Y)\leq 3\epsilon/2$. Conversely, if
$d_{{\rm GH}}(X,Y)<\epsilon$, then there exists a $2\epsilon$-Hausdorff
approximation map from $X$ to $Y$.

######  Definition 2.15.

Suppose that a sequence of $n$-dimensional closed Riemannian manifolds
$\\{(M_{i},g_{i})\\}_{i\in\mathbb{Z}_{>0}}$ with $\operatorname{Ric}_{i}\geq
Kg_{i}$ and $\operatorname{diam}(M_{i})\leq D$ ($K\in\mathbb{R}$, $D>0$)
converges to a metric space $(X,d)$ in the Gromov–Hausdorff topology. Fix a
sequence of $\epsilon_{i}$-Hausdorff approximation maps $\psi_{i}\colon
M_{i}\to X$, where $\\{\epsilon_{i}\\}$ is some sequence of positive real
numbers with $\lim\limits_{i\to\infty}\epsilon_{i}=0$.

  * •

We say a sequence $\\{x_{i}\\}$ with $x_{i}\in M_{i}$ converges to $x\in X$ if
$\lim\limits_{i\to\infty}\psi_{i}(x_{i})=x$ in $X$ $\big{(}$denote it by
$x_{i}\stackrel{{\scriptstyle{\rm GH}}}{{\to}}x\big{)}$.

  * •

Let $\mathfrak{m}$ be a Borel measure on $X$. We say that a sequence
$\\{(M_{i},g_{i},\underline{\mathcal{H}}^{n})\\}$ converges to a metric
measure space $(X,d,\mathfrak{m})$ in the measured Gromov–Hausdorff topology
if

$\lim_{i\to\infty}\underline{\mathcal{H}}^{n}(B_{r}(x_{i}))=\mathfrak{m}(B_{r}(x))$

holds for any $r>0$, $x_{i}\in M_{i}$ and $x\in X$ with
$x_{i}\stackrel{{\scriptstyle{\rm GH}}}{{\to}}x$. Note that taking a
subsequence, such a limit measure exists by [12, Theorems 1.6 and 1.10].
Moreover, $(X,d,\mathfrak{m})$ satisfies the ${\rm RCD}^{\ast}(K,N)$ condition
by [14, Theorem 3.22].

Suppose that a sequence $\\{(M_{i},g_{i},\underline{\mathcal{H}}^{n})\\}$
converges to $(X,d,\mathfrak{m})$ in the measured Gromov–Hausdorff topology.

  * •

We say that $f_{i}\in L^{2}(M_{i})$ ($i\in\mathbb{Z}_{>0}$) converges to $f\in
L^{2}(X)$ strongly at $x\in X$ [22, Definition 3.7] if we have that

$\displaystyle\lim_{r\to
0}\limsup_{i\to\infty}\bigg{(}\frac{1}{\underline{\mathcal{H}}^{n}(B_{r}(x_{i}))}\int_{B_{r}(x_{i})}\bigg{|}f_{i}-\frac{1}{\mathfrak{m}(B_{r}(x))}\int_{B_{r}(x)}f\,{\rm
d}\mathfrak{m}\bigg{|}\,{\rm d}\underline{\mathcal{H}}^{n}\bigg{)}=0,$
$\displaystyle\lim_{r\to
0}\limsup_{i\to\infty}\bigg{(}\frac{1}{\mathfrak{m}(B_{r}(x))}\int_{B_{r}(x)}\bigg{|}f-\frac{1}{\underline{\mathcal{H}}^{n}(B_{r}(x_{i}))}\int_{B_{r}(x_{i})}f_{i}\,{\rm
d}\underline{\mathcal{H}}^{n}\bigg{|}\,{\rm d}\mathfrak{m}\bigg{)}=0$

for all $x_{i}\in M_{i}$ with $x_{i}\stackrel{{\scriptstyle{\rm
GH}}}{{\to}}x$.

  * •

We say that $f_{i}\in L^{2}(M_{i})$ ($i\in\mathbb{Z}_{>0}$) converges to $f\in
L^{2}(X)$ weakly in $L^{2}$ ($L^{2}$ boundedness and weakly convergence [22,
Definition 3.4 and Proposition 3.17]) if

$\displaystyle\sup_{i}\|f_{i}\|_{L^{2}}<\infty,$

and for all $r>0$, $x_{i}\in M_{i}$ and $x\in X$ with
$x_{i}\stackrel{{\scriptstyle{\rm GH}}}{{\to}}x$, we have

$\displaystyle\lim_{i\to\infty}\int_{B_{r}(x_{i})}f_{i}\,{\rm
d}\underline{\mathcal{H}}^{n}=\int_{B_{r}(x)}f\,{\rm d}\mathfrak{m}.$

Note that we have
$\|f\|_{L^{2}}\leq\liminf\limits_{i\to\infty}\|f_{i}\|_{L^{2}}$ by [22,
Proposition 3.29].

  * •

We say that $f_{i}\in L^{2}(M_{i})$ ($i\in\mathbb{Z}_{>0}$) converges to $f\in
L^{2}(X)$ strongly in $L^{2}$ [22, Definition 3.21 and Proposition 3.31] if
$f_{i}$ converges to $f$ weakly in $L^{2}$, and

$\displaystyle\limsup_{i\to\infty}\|f_{i}\|_{L^{2}}\leq\|f\|_{L^{2}}$

holds.

  * •

We say that $V_{i}\in L^{2}(TM_{i})$ ($i\in\mathbb{Z}_{>0}$) converges to
$V\in L^{2}(TX)$ weakly in $L^{2}$ [22, Definition 3.42] if

$\displaystyle\sup_{i}\|V_{i}\|_{L^{2}}<\infty,$

and for all $r>0$, $y_{i},z_{i}\in X_{i}$ and $y,z\in X$ with
$y_{i}\stackrel{{\scriptstyle{\rm GH}}}{{\to}}y$,
$z_{i}\stackrel{{\scriptstyle{\rm GH}}}{{\to}}z$, we have

$\displaystyle\lim_{i\to\infty}\int_{B_{r}(y_{i})}\langle V_{i},\nabla
r_{z_{i}}\rangle\,{\rm d}\underline{\mathcal{H}}^{n}=\int_{B_{r}(y)}\langle
V,\nabla r_{z}\rangle\,{\rm d}\mathfrak{m},$

where $r_{z}(x):=d(z,x)$ for each $x\in X$. Note that we have
$\|V\|_{L^{2}}\leq\liminf\limits_{i\to\infty}\|V_{i}\|_{L^{2}}$ by [22,
Proposition 3.64].

  * •

We say that $V_{i}\in L^{2}(TM_{i})$ ($i\in\mathbb{Z}_{>0}$) converges to
$X\in L^{2}(TX)$ strongly in $L^{2}$ [22, Definition 3.58 and Proposition
3.66] if $V_{i}$ converges to $V$ weakly in $L^{2}$ and

$\displaystyle\limsup_{i\to\infty}\|V_{i}\|_{2}\leq\|V\|_{2}$

holds.

######  Proposition 2.16 ([22, Proposition 3.32]).

Suppose that a sequence of $n$-dimensional closed Riemannian manifolds
$\\{(M_{i},g_{i},\underline{\mathcal{H}}^{n})\\}_{i\in\mathbb{Z}_{>0}}$ with
$\operatorname{Ric}_{i}\geq Kg_{i}$ and $\operatorname{diam}(M_{i})\leq D$
$(K\in\mathbb{R}$, $D>0)$ converges to a compact metric measure space
$(X,d,\mathfrak{m})$ in the measured Gromov–Hausdorff topology. For any
$f_{i}\in L^{\infty}(M_{i})$ $(i\in\mathbb{Z}_{>0})$ with
$\sup\|f_{i}\|_{L^{\infty}}<\infty$ and $f\in L^{\infty}(X)$, the following
conditions are mutually equivalent:

  * $(i)$

$f_{i}\to f$ strongly at a.e. $x\in X$.

  * $(ii)$

$f_{i}\to f$ strongly in $L^{2}$.

Note that the implication $(i)\Rightarrow(ii)$ is a direct consequence of [22,
Definitions 3.21 and 3.25, Proposition 3.24].

######  Theorem 2.17 ([22, Theorem 1.3]).

Suppose that a sequence of $n$-dimensional closed Riemannian manifolds
$\\{(M_{i},g_{i},\underline{\mathcal{H}}^{n})\\}_{i\in\mathbb{Z}_{>0}}$ with
$\operatorname{Ric}_{i}\geq Kg_{i}$ and $\operatorname{diam}(M_{i})\leq D$
$(K\in\mathbb{R}$, $D>0)$ converges to a compact metric measure space
$(X,d,\mathfrak{m})$ in the measured Gromov–Hausdorff topology. If $f_{i}\in
L^{2}(M_{i})\cap C^{2}(M_{i})$ $(i\in\mathbb{Z}_{>0})$ converges to $f\in
L^{2}(X)$ weakly in $L^{2}$, and satisfies

$\sup_{i\in\mathbb{Z}_{>0}}\left(\|f_{i}\|_{W^{1,2}}+\|\Delta
f_{i}\|_{L^{2}}\right)<\infty,$

then we have the following:

  * $(i)$

$f_{i}\to f$ and $\nabla f_{i}\to\nabla f$ strongly in $L^{2}$,

  * $(ii)$

$f\in\mathcal{D}(\Delta_{X})$ and $\Delta f_{i}\to\Delta f$ weakly in $L^{2}$.

###  2.5 Convergence to the product space $\boldsymbol{S^{n-p}(1)\times X}$

We summarize several results proven in [1] for later use.

######  Proposition 2.18 ([1, Proposition 4.17, Lemma 4.22 and Theorem 4.47]).

For given integers $n\geq 5$ and $2\leq p<n/2$ and a positive real number
$\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that the
following properties hold. Let $(M,g)$ be an $n$-dimensional closed Riemannian
manifold with $\operatorname{Ric}_{g}\geq(n-p-1)g$. Assume that
$\lambda_{n-p+1}(g)\leq n-p+\delta$ and that either
$\lambda_{1}(\Delta_{C,p})\leq\delta$ or
$\lambda_{1}(\Delta_{C,n-p})\leq\delta$. Then, for any
$f\in\operatorname{Span}_{\mathbb{R}}\\{f_{1},\ldots,f_{n-p+1}\\}$ with
$\|f\|_{2}^{2}=1/(n-p+1)$, we have the following:

  * $(i)$

There exists a measurable subset $V_{f}\subset M$ such that
$\operatorname{Vol}(M\setminus V_{f})\leq\epsilon\operatorname{Vol}(M)$ and
$|f^{2}+|\nabla f|^{2}-1|\leq\epsilon$ holds in $V_{f}$.

  * $(ii)$

There exists a non-empty compact subset $A_{f}\subset M$ such that
$|f(x)-1|\leq\epsilon$ for any $x\in A_{f}$, $|f(x)-\cos
d(x,A_{f})|\leq\epsilon$ for any $x\in M$ and $\sup_{x\in
M}d(x,A_{f})\leq\pi+\epsilon$ hold.

  * $(iii)$

Define $\widetilde{\Psi}\colon M\to\mathbb{R}^{n-p+1}$ by
$\widetilde{\Psi}(x)=(f_{1}(x),\ldots,f_{n-p+1}(x))$ $(x\in M)$. Then, we have
$||\widetilde{\Psi}(x)|-1|\leq\epsilon$ for any $x\in M$.

  * $(iv)$

Choose $a_{f}(x)\in A_{f}$ such that $d(x,A_{f})=d(x,a_{f}(x))$ for each $x\in
M$. Then, we have that the map

$\Phi_{f}\colon\ M\to S^{n-p}(1)\times A_{f},\qquad
x\mapsto\bigg{(}\frac{\widetilde{\Psi}(x)}{|\widetilde{\Psi}(x)|},a_{f}(x)\bigg{)}$

is an $\epsilon$-Hausdorff approximation.

######  Proposition 2.19 ([1, Corollary 4.53]).

For given integers $n\geq 5$ and $2\leq p<n/2$ and a positive real number
$\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that if
$(M,g)$ is an $n$-dimensional closed Riemannian manifold with
$\operatorname{Ric}_{g}\geq(n-p-1)g$, $\lambda_{n-p}(g)\leq n-p+\delta$ and
$\lambda_{1}(\Delta_{C,n-p})\leq\delta,$ then we have $\lambda_{n-p+1}(g)\leq
n-p+\epsilon.$

If we assume $\lambda_{1}(\Delta_{C,p})\leq\delta$ instead of
$\lambda_{1}(\Delta_{C,n-p})\leq\delta$ in Proposition 2.19, the assertion
fails (see [1, Corollary 3.2 and Proposition 3.3]).

##  3 Structure of the limit

###  3.1 Splitting of the measure

In this subsection we show that there exists a Borel measure
$\mathfrak{m}_{X}$ on X such that
$\mathfrak{m}=\underline{\mathcal{H}}^{n-p}\times\mathfrak{m}_{X}$ holds under
Assumption 3.1 below.

######  Assumption 3.1.

Take $n\geq 5$ and $2\leq p<n/2$. Let $\\{(M_{i},g_{i})\\}_{i\in\mathbb{N}}$
be a sequence of $n$-dimensional closed Riemannian manifolds with
$\operatorname{Ric}_{g_{i}}\geq(n-p-1)g_{i}$ that satisfies
$\lim\limits_{i\to\infty}\lambda_{n-p+1}(g_{i})=n-p$ and either
$\lim\limits_{i\to\infty}\lambda_{1}(\Delta_{C,p},g_{i})=0$ or
$\lim\limits_{i\to\infty}\lambda_{1}(\Delta_{C,n-p},g_{i})=0$. Let
$f_{1,i},\ldots,f_{n-p+1,i}$ denote the first $n-p+1$ eigenfunctions on
$(M_{i},g_{i})$ with $\|f_{k,i}\|_{L^{2}}^{2}=1/(n-p+1)$ ($k=1,\ldots,n-p+1$).
Put

$\Psi_{i}\colon\ M_{i}\to S^{n-p}(1),\qquad
x\mapsto\frac{(f_{1,i}(x),\ldots,f_{n-p+1,i}(x))}{|(f_{1,i}(x),\ldots,f_{n-p+1,i}(x))|}.$

Let $X$ be a compact metric space and $\mathfrak{m}$ be a Borel measure on
$S^{n-p}(1)\times X$ with unit volume. Suppose that, for each $i$, there
exists a map $b_{i}\colon M_{i}\to X$ such that the map

$(\Psi_{i},b_{i})\colon\ M_{i}\to S^{n-p}(1)\times X$

is an $\epsilon_{i}$-approximation, where $\\{\epsilon_{i}\\}$ is some
sequence of positive real numbers with $\epsilon_{i}\to 0$ as $i\to\infty$.
Suppose that the sequence $\\{(M_{i},g_{i},\underline{\mathcal{H}}^{n})\\}$
converges to $(S^{n-p}(1)\times X,\mathfrak{m})$ in the measured
Gromov–Hausdorff topology. Put $M:=S^{n-p}(1)\times X$. Let $p_{1}\colon M\to
S^{n-p}(1)$ and $p_{2}\colon M\to X$ be the projections. Define $f_{u}\colon
M\to\mathbb{R}$ by $f_{u}(x):=p_{1}(x)\cdot u$ for each $u\in
S^{n-p}(1)\subset\mathbb{R}^{n-p+1}$. Note that $(M,\mathfrak{m})$ satisfies
the ${\rm RCD}^{\ast}(n-p-1,n)$ condition by [14, Theorem 3.22].

The goal of this section is to prove the following proposition.

######  Proposition 3.2.

Under Assumption 3.1, there exists a Borel measure $\mathfrak{m}_{X}$ on X
such that $\mathfrak{m}=\underline{\mathcal{H}}^{n-p}\times\mathfrak{m}_{X}$
holds and $(X,\mathfrak{m}_{X})$ satisfies the $\mathrm{RCD}(n-p-1,p)$
condition.

In this subsection, we show the splitting of the measure
$\mathfrak{m}=\underline{\mathcal{H}}^{n-p}\times\mathfrak{m}_{X}$. Our
approach has been inspired by [17]. We first show the following easy lemma.

######  Lemma 3.3.

Define

$f_{k}\colon\ S^{n-p}(1)\times
X\to\mathbb{R},\,((u_{1},\ldots,u_{n-p+1}),x)\mapsto u_{k}$

for each $k=1,\ldots,n-p+1$. Then, we have that
$f_{k}\in\operatorname{TestF}(M)$ and the following properties:

  * $(i)$

For each $k=1,\ldots,n-p+1$, the sequence $\\{f_{k,i}\\}$ converges to $f_{k}$
strongly in $L^{2}$ as $i\to\infty$.

  * $(ii)$

For each $k=1,\ldots,n-p+1$, the sequence $\\{\nabla f_{k,i}\\}$ converges to
$\nabla f_{k}$ strongly in $L^{2}$ as $i\to\infty$.

  * $(iii)$

For each $k=1,\ldots,n-p+1$, we have that $\Delta f_{k}=-(n-p)f_{k}$.

  * $(iv)$

For each $k,l=1,\ldots,n-p+1$ with $k\neq l$, we have that

$\displaystyle f_{k}f_{l}+\langle\nabla f_{k},\nabla f_{l}\rangle=0,$
$\displaystyle f_{k}^{2}+|\nabla f_{k}|^{2}=1$

$\mathfrak{m}$-a.e. in $M$.

###### Proof.

Clearly, $f_{k}$ is a Lipschitz function. If we get $(iii)$, we have $\Delta
f_{k}=-(n-p)f_{k}\in W^{1,2}(M)$, and so we have
$f_{k}\in\operatorname{TestF}(M)$.

We first show that $\\{f_{k,i}\\}$ strongly converges to $f_{k}$ as
$i\to\infty$ at each point $z\in M$. Note that by the gradient estimate for
eigenfunctions [27, Theorem 7.3], there exists a constant $C>0$ such that
$\|f_{k,i}\|_{L^{\infty}}+\|\nabla f_{k,i}\|_{L^{\infty}}\leq C$ holds for all
$i\in\mathbb{Z}_{>0}$ and $k=1,\ldots,n-p+1$. Take arbitrary $z=(u,x)\in
S^{n-p}(1)\times X$ and $z_{i}\in M_{i}$ ($i\in\mathbb{Z}_{>0}$) with
$z_{i}\stackrel{{\scriptstyle{\rm GH}}}{{\to}}z$. Since we have
$\Psi_{i}(z_{i})\to u$ in $S^{n-p}(1)$, we have that

$\lim_{i\to\infty}\bigg{|}\frac{f_{k,i}(z_{i})}{|(f_{1,i},\ldots,f_{n-p+1,i})|(z_{i})}-f_{k}(z)\bigg{|}=0,$

and so $\lim_{i\to\infty}|f_{k,i}(z_{i})-f_{k}(z)|=0$ by Proposition
2.18$(iii)$. Since $f_{k,i}$ and $f_{k}$ are Lipschitz functions whose
Lipschitz constants are bounded independently of $i$, we have that

$\displaystyle\frac{1}{\underline{\mathcal{H}}^{n}(B_{r}(z_{i}))}\int_{B_{r}(z_{i})}\bigg{|}f_{k,i}-\frac{1}{\mathfrak{m}(B_{r}(z))}\int_{B_{r}(z)}f_{k}\,{\rm
d}\mathfrak{m}\bigg{|}\,{\rm
d}\underline{\mathcal{H}}^{n}\leq\frac{1}{\underline{\mathcal{H}}^{n}(B_{r}(z_{i}))}$
$\displaystyle\qquad\qquad\times\int_{B_{r}(z_{i})}\\!\\!\bigg{(}\\!|f_{k,i}\\!-\\!f_{k,i}(z_{i})|\\!+\\!|f_{k,i}(z_{i})\\!-\\!f_{k}(z)|+\frac{1}{\mathfrak{m}(B_{r}(z))}\\!\int_{B_{r}(z)}\\!\\!|f_{k}-f_{k}(z)|\,{\rm
d}\mathfrak{m}\\!\bigg{)}{\rm d}\underline{\mathcal{H}}^{n}$
$\displaystyle\qquad\leq Cr+|f_{k,i}(z_{i})-f_{k}(z)|.$

Thus,

$\displaystyle\lim_{r\to
0}\limsup_{i\to\infty}\frac{1}{\underline{\mathcal{H}}^{n}(B_{r}(z_{i}))}\int_{B_{r}(z_{i})}\bigg{|}f_{k,i}-\frac{1}{\mathfrak{m}(B_{r}(z))}\\!\int_{B_{r}(z)}f_{k}\,{\rm
d}\mathfrak{m}\bigg{|}\,{\rm d}\underline{\mathcal{H}}^{n}=0.$

Similarly, we have

$\lim_{r\to
0}\limsup_{i\to\infty}\frac{1}{\mathfrak{m}^{n}(B_{r}(z))}\int_{B_{r}(z)}\bigg{|}f_{k}-\frac{1}{\underline{\mathcal{H}}^{n}(B_{r}(z_{i}))}\int_{B_{r}(z_{i})}f_{k,i}\,{\rm
d}\underline{\mathcal{H}}^{n}\bigg{|}\,{\rm d}\mathfrak{m}=0.$

Therefore, we get $(i)$ by Proposition 2.16. We get $(ii)$ by Theorem
2.17$(i)$. By Theorem 2.17$(ii)$, we have

$\|\Delta f_{k}+(n-p)f_{k}\|_{L^{2}}\leq\liminf_{i\to\infty}\|\Delta
f_{k,i}+(n-p)f_{k,i}\|_{L^{2}}=0,$

and so we get $(iii)$. For each $k=1,\ldots,n-p+1$, we have that

$\|f_{k}^{2}+|\nabla
f_{k}|^{2}-1\|_{L^{1}}=\lim_{i\to\infty}\|f_{k,i}^{2}+|\nabla
f_{k,i}|^{2}-1\|_{L^{1}},$

by [22, Propositions 3.11 and 3.45] and the original definition of the $L^{2}$
strong convergence [22, Definitions 3.25 and 3.58]. See also [5, Proposition
3.3 and Theorem 5.7]. By Proposition 2.18$(i)$, we have
$\lim\limits_{i\to\infty}\|f_{k,i}^{2}+|\nabla f_{k,i}|^{2}-1\|_{L^{1}}=0$,
and so we get $f_{k}^{2}+|\nabla f_{k}|^{2}=1$ $\mathfrak{m}$-a.e. in $M$.
Similarly, applying Proposition 2.18$(i)$ to $(f_{k,i}\pm
f_{l,i})/{\sqrt{2}}$, we get $f_{k}f_{l}+\langle\nabla f_{k},\nabla
f_{l}\rangle=0$ $\mathfrak{m}$-a.e. in $M$ for each $k,l=1,\ldots,n-p+1$ with
$k\neq l$. These imply $(iv)$. ∎

Let us apply Theorem 2.10 to vector fields generating rotations in
$S^{n-p}(1)$.

######  Lemma 3.4.

Take arbitrary $u=(u_{1},\ldots,u_{n-p+1}),v=(v_{1},\ldots,v_{n-p+1})\in
S^{n-p}(1)$ with $u\cdot v=0$ and $T>0$. Then, the vector field

$V_{uv}:=\sum_{i,j=1}^{n-p+1}u_{i}v_{j}(f_{i}\nabla f_{j}-f_{j}\nabla
f_{i})=f_{u}\nabla f_{v}-f_{v}\nabla f_{u}$

is an element of $\mathcal{D}(\operatorname{div})$ with
$D^{\operatorname{sym}}V_{uv}\in L^{2}(M)$ and $\operatorname{div}V_{uv}=0$.
Moreover, the regular Lagrangian flow $\operatorname{Fl}^{V_{uv}}\colon
M\times[0,T]\to M$ for $V_{uv}$ exists and satisfies, for $\mathfrak{m}$-a.e.
$z\in M$,

$\displaystyle f_{u}(\operatorname{Fl}^{V_{uv}}_{t}(z))=f_{u}(z)\cos
t-f_{v}(z)\sin t,$ $\displaystyle
f_{v}(\operatorname{Fl}^{V_{uv}}_{t}(z))=f_{u}(z)\sin t+f_{v}(z)\cos t,$
$\displaystyle f_{w}(\operatorname{Fl}^{V_{uv}}_{t}(z))=f_{w}(z)\qquad(w\in
S^{n-p}(1)\text{ with }u\cdot w=v\cdot w=0),$ $\displaystyle
p_{2}(\operatorname{Fl}^{V_{uv}}_{t}(z))=p_{2}(z)$

for any $t\in[0,T]$. Moreover, $\operatorname{Fl}^{V_{uv}}$ preserves the
measure $\mathfrak{m}$, i.e.,

$\big{(}\operatorname{Fl}^{V_{uv}}_{t}\big{)}_{\ast}\mathfrak{m}=\mathfrak{m}$

for any $t\in[0,T]$.

###### Proof.

Since we have

$\operatorname{div}(f_{k}\nabla f_{l})=-(n-p)f_{k}f_{l}+\langle\nabla
f_{k},\nabla f_{l}\rangle$

for each $k$, $l$ by (2.3.13) in [15], we get $\operatorname{div}(V_{uv})=0$.
We next check $D^{\operatorname{sym}}V_{uv}\in L^{2}(X)$. It is enough to show
(2.1) when $f,g\in\operatorname{TestF}(M)$ because we have $\|\nabla f-\nabla
P_{t}f\|_{L^{4}}\to 0$ and $\|\Delta f-\Delta P_{t}f\|_{L^{4}}\to 0$ as $t\to
0$ for each $f\in\mathcal{D}(\Delta)$ with $|\nabla f|\in L^{4}(M)$ and
$\Delta f\in L^{4}(M)$. Note that

$L^{4}(TM):=\\{V\in L^{2}(TM)\colon|V|\in L^{4}(M)\\}$

is a uniformly convex Banach space with the norm
$\|V\|_{L^{4}}:=\||V|\|_{L^{4}}$, and that $\|\nabla
P_{t}f\|_{L^{4}}\leq\|\nabla f\|_{L^{4}}$ for each $f\in W^{1,2}(X)$ with
$|\nabla f|\in L^{4}(X)$ by the ${\rm BL}(n-p-1,n)$ condition for
$(M,\mathfrak{m})$. Combining these and $\nabla P_{t}f\to\nabla f$ in
$L^{2}(TM)$, we get $\nabla P_{t}f\to\nabla f$ in $L^{4}(TM)$.

Take $f,g\in\operatorname{TestF}(M)$. A simple calculation implies

$\displaystyle\int_{M}\langle V_{uv},\nabla f\rangle\Delta g+\langle
V_{uv},\nabla g\rangle\Delta f-\langle\nabla f,\nabla
g\rangle\operatorname{div}V_{uv}\,{\rm d}\mathfrak{m}$
$\displaystyle\qquad{}=\int_{M}\langle\nabla f_{v},\nabla
f\rangle\operatorname{div}(f_{u}\nabla g)+\langle\nabla f_{v},\nabla
g\rangle\operatorname{div}(f_{u}\nabla f)+f_{u}\langle\nabla
f_{v},\nabla\langle\nabla f,\nabla g\rangle\rangle\,{\rm d}\mathfrak{m}$
$\displaystyle\phantom{qquad=}-\int_{M}\langle\nabla f_{u},\nabla
f\rangle\operatorname{div}(f_{v}\nabla g)+\langle\nabla f_{u},\nabla
g\rangle\operatorname{div}(f_{v}\nabla f)+f_{v}\langle\nabla
f_{u},\nabla\langle\nabla f,\nabla g\rangle\rangle\,{\rm d}\mathfrak{m}.$

Note that we have $\langle\nabla f,\nabla g\rangle\in W^{1,2}(M)$ by [15,
Proposition 3.1.3]. By [15, Definition 3.3.1 and Theorem 3.3.8], we get
$D^{\operatorname{sym}}V_{uv}\in L^{2}(M)$. Thus, there exists a regular
Lagrangian flow $\operatorname{Fl}^{V_{uv}}\colon M\times[0,T]\to M$ for
$V_{uv}$ by Theorem 2.10.

For $\mathfrak{m}$-a.e. $z\in M$, we have

$\displaystyle\frac{\rm d}{{\rm
d}t}f_{w}\left(\operatorname{Fl}^{V_{uv}}(t,z)\right)=$
$\displaystyle\left(f_{u}\langle\nabla f_{v},\nabla f_{w}\rangle-
f_{v}\langle\nabla f_{u},\nabla
f_{w}\rangle\right)\left(\operatorname{Fl}^{V_{uv}}(t,z)\right)$
$\displaystyle=$
$\displaystyle\begin{cases}-f_{v}\big{(}\operatorname{Fl}^{V_{uv}}(t,z)\big{)}&(w=u),\\\
f_{u}\big{(}\operatorname{Fl}^{V_{uv}}(t,z)\big{)}&(w=v),\\\ 0&(w\in
S^{n-p}(1)\text{ with }u\cdot w=v\cdot w=0)\end{cases}$

for a.e. $t\in(0,T)$ by Lemma 3.3. This implies that for $\mathfrak{m}$-a.e.
$z\in M$,

$f_{w}\left(\operatorname{Fl}^{V_{uv}}(t,z)\right)=\begin{cases}f_{u}(z)\cos
t-f_{v}(z)\sin t&(w=u),\\\ f_{u}(z)\sin t+f_{v}(z)\cos t&(w=v),\\\
f_{w}(z)&(w\in S^{n-p}(1)\text{ with }u\cdot w=v\cdot w=0)\end{cases}$

for any $t\in[0,T]$.

A simple calculation implies that

$\langle\nabla f_{k},\nabla(g\circ p_{2})\rangle=0$

$\mathfrak{m}$-a.e. in $M$ for each $g\in\operatorname{LIP}(X)$ and
$k=1,\ldots,n-p+1$ similarly to Lemma 3.11 $(iv)$ below. Therefore, for each
$g\in\operatorname{LIP}(X)$ and $\mathfrak{m}$-a.e. $z\in M$, we have

$\frac{\rm d}{{\rm d}t}(g\circ
p_{2})\left(\operatorname{Fl}^{V_{uv}}(t,z)\right)=0$

for a.e. $t\in[0,T]$, and so

$(g\circ p_{2})\left(\operatorname{Fl}^{V_{uv}}(t,z)\right)=g\circ p_{2}(z)$

for any $t\in[0,T]$. Let $\\{x_{j}\\}_{j\in\mathbb{Z}_{>0}}$ be a countable
dense subset of $X$. Then, by considering $g_{j}:=d(x_{j},\cdot)$, we get that
for $\mathfrak{m}$-a.e. $z\in M$,

${\rm d}\big{(}x_{j},p_{2}(\operatorname{Fl}^{V_{uv}}(t,z))\big{)}={\rm
d}(x_{j},p_{2}(z))$

holds for any $j\in\mathbb{Z}_{>0}$ and $t\in[0,T]$. This implies for
$\mathfrak{m}$-a.e. $z\in M$,
$p_{2}\left(\operatorname{Fl}^{V_{uv}}(t,z)\right)=p_{2}(z)$ for any
$t\in[0,T]$.

We have that
$\operatorname{Fl}^{V_{uv}}_{t}\circ\operatorname{Fl}_{t}^{-V_{uv}}=\operatorname{Id}_{M}$
$\mathfrak{m}$-a.e. in $M$ for all $t\in[0,T]$ (note that $-V_{uv}=V_{vu}$),
and so

$\mathfrak{m}=\big{(}\operatorname{Fl}^{V_{uv}}_{t}\big{)}_{\ast}\big{(}\operatorname{Fl}_{t}^{-V_{uv}}\big{)}_{\ast}\mathfrak{m}\leq\big{(}\operatorname{Fl}_{t}^{V_{uv}}\big{)}_{\ast}\mathfrak{m}\leq\mathfrak{m}.$

This implies the final assertion. ∎

######  Corollary 3.5.

For any $T\in\mathrm{SO}(n-p+1)$, the transformation

$T\colon\ S^{n-p}(1)\times X\to S^{n-p}(1)\times X,\,(u,x)\mapsto(Tu,x)$

preserves the measure $\mathfrak{m}$.

###### Proof.

Modifying on $\mathfrak{m}$-negligible subset, we have that
$\operatorname{Fl}^{V_{uv}}_{t}\in\mathrm{SO}(n-p+1)$ for each $u,v\in
S^{n-p}(1)$ with $u\cdot v=0$ and $t\in[0,2\pi]$. Conversely, any
$T\in\mathrm{SO}(n-p+1)$ can be expressed as a composition of several
transformations of the form $\operatorname{Fl}^{V_{uv}}_{t}$. Thus, we get the
corollary. ∎

The following proposition is the goal of this subsection.

######  Proposition 3.6.

Define a Borel measure $\mathfrak{m}_{X}$ on $X$ by

$\mathfrak{m}_{X}:=(p_{2})_{\ast}\mathfrak{m}.$

Then, we have
$\mathfrak{m}=\underline{\mathcal{H}}^{n-p}\times\mathfrak{m}_{X}$.

###### Proof.

We first fix a Borel subset $B\subset X$. Define a Borel measure $\mu_{B}$ on
$S^{n-p}(1)$ by

$\mu_{B}:=(p_{1})_{\ast}\big{(}\mathfrak{m}|_{p_{2}^{-1}(B)}\big{)},$

i.e., we define

$\mu_{B}(A):=\mathfrak{m}(A\times B)$

for any subset $A\subset S^{n-p}(1)$.

######  Claim 3.7.

Each $T\in\mathrm{SO}(n-p+1)$ preserves the measure $\mu_{B}$.

###### Proof.

We immediately have the claim by Corollary 3.5. ∎

######  Claim 3.8.

$\mu_{B}\ll\mathcal{H}^{n-p}$.

###### Proof.

By the volume estimate relative to $\mathcal{H}^{n-p}$ on $S^{n-p}(1)$, there
exists a constant $C>0$ such that

$\max\left\\{k\in\mathbb{Z}_{>0}\colon\begin{array}[]{@{}l@{}}\text{there
exist $x_{1},\ldots,x_{k}\in S^{n-p}(1)$ such that}\\\ \text{$B_{r}(x_{i})\cap
B_{r}(x_{j})=\varnothing$ holds for each $i\neq j$}\end{array}\right\\}\geq
r^{-(n-p)}/C$

holds for all $r>0$.

Take $r>0$. We can choose $k\in\mathbb{Z}_{>0}$ with $k\geq r^{-(n-p)}/C$ and
$x_{1},\ldots,x_{k}\in S^{n-p}(1)$ such that $B_{r}(x_{i})\cap
B_{r}(x_{j})=\varnothing$ holds for each $i\neq j$. By Claim 3.7, we have that

$\mu_{B}(B_{r}(x_{i}))=\mu_{B}(B_{r}(x_{j}))=\mu_{B}(B_{r}(x))$

for all $i,j=1,\ldots,k$ and $x\in S^{n-p}(1)$. Therefore, we get that
$\mu_{B}(B_{r}(x))\leq Cr^{n-p}\mu_{B}(S^{n-p}(1))$ for all $x\in S^{n-p}(1)$.

Take arbitrary subset $A\subset S^{n-p}(1)$ with $\mathcal{H}^{n-p}(A)=0$ and
$\epsilon>0$. Then, by the definition of the Hausdorff measure, there exists a
sequence of subsets $\\{S_{j}\\}_{j\in\mathbb{Z}_{>0}}$ of $S^{n-p}(1)$ such
that $A\subset\bigcup_{j=1}^{\infty}S_{j}$ and

$\sum_{j=1}^{\infty}(\operatorname{diam}S_{j})^{n-p}<\epsilon.$

Choose $x_{j}\in S_{j}$ for each $j$. Then, we have $S_{j}\subset
B_{\operatorname{diam}S_{j}}(x_{j})$, and so

$\mu_{B}(A)\leq\sum_{j=1}^{\infty}\mu_{B}(B_{\operatorname{diam}S_{j}}(x_{j}))\leq
C\sum_{j=1}^{\infty}(\operatorname{diam}S_{j})^{n-p}\mu_{B}(S^{n-p}(1))\leq
C\epsilon\mu_{B}\big{(}S^{n-p}(1)\big{)}.$

Letting $\epsilon\to 0$, we obtain $\mu_{B}(A)=0$ and get the claim. ∎

By Claim 3.8 and the Radon–Nikodym theorem, we have the representation
$\mu_{B}=\rho\mathcal{H}^{n-p}$, where $\rho\colon S^{n-p}(1)\to[0,\infty]$ is
some Borel function. By Claim 3.7, we have that for each
$T\in\mathrm{SO}(n-p+1)$

$\rho\circ T=\rho$

$\mathcal{H}^{n-p}$-a.e. in $S^{n-p}(1)$. This implies that $\rho$ is constant
$\mathcal{H}^{n-p}$-a.e. in $S^{n-p}(1)$. We have that

$\rho\mathcal{H}^{n-p}\big{(}S^{n-p}(1)\big{)}=\mu_{B}\big{(}S^{n-p}(1)\big{)}=\mathfrak{m}\big{(}S^{n-p}(1)\times
B\big{)}=\mathfrak{m}_{X}(B),$

and so $\mu_{B}=\mathfrak{m}_{X}(B)\underline{\mathcal{H}}^{n-p}$.

For each Borel sets $A\subset S^{n-p}(1)$ and $B\subset X$, we get that

$\mathfrak{m}(A\times
B)=\mu_{B}(A)=\underline{\mathcal{H}}^{n-p}(A)\mathfrak{m}_{X}(B).$

This implies the proposition.

###  3.2 Product metric measure spaces and the RCD∗ condition

In the previous subsection we showed that there exists a Borel measure
$\mathfrak{m}_{X}$ on X such that
$\mathfrak{m}=\underline{\mathcal{H}}^{n-p}\times\mathfrak{m}_{X}$ holds under
Assumption 3.1. In this subsection we show that $(X,\mathfrak{m}_{X})$
satisfies the $\mathrm{RCD}(n-p-1,p)$ condition.

More generally, we consider the following assumption.

######  Assumption 3.9.

Let $K,N\in\mathbb{R}$ with $N\geq 1$ and $(X_{i},d_{i},\mathfrak{m}_{i})$
($i=1,2$) be compact metric measure spaces. Put
$(M,d,\mathfrak{m}):=(X_{1}\times X_{2},d_{1}\times
d_{2},\mathfrak{m}_{1}\times\mathfrak{m}_{2})$. Moreover, we assume the
following:

  * •

$\mathfrak{m}_{i}(X_{i})=1$ ($i=1,2$),

  * •

$(M,d,\mathfrak{m})$ satisfies the ${\rm RCD}^{\ast}(K,N)$ condition.

For each $i=1,2$, let $p_{i}\colon M\to X_{i}$ denote the projection.

The goal of this subsection is to prove the following proposition:

######  Proposition 3.10.

In addition to Assumption 3.9, we assume that $(X_{1},d_{1},\mathfrak{m}_{1})$
is an $n$-dimensional closed Riemannian manifold with the Riemannian distance
and $\mathfrak{m}_{1}=\underline{\mathcal{H}}^{n}$. Then,
$(X_{2},d_{2},\mathfrak{m}_{2})$ satisfies the ${\rm RCD}^{\ast}(K,N-n)$
condition if $N-n\geq 1$.

Note that if $(Y_{i},d^{\prime}_{i},\mathfrak{m}^{\prime}_{i})$ ($i=1,2$) are
${\rm RCD}^{\ast}(K,N_{i})$ spaces, then the product space $(Y_{1}\times
Y_{2},\allowbreak d^{\prime}_{1}\times
d^{\prime}_{2},\mathfrak{m}^{\prime}_{1}\times\mathfrak{m}_{2}^{\prime})$
satisfies the ${\rm RCD}^{\ast}(K,N_{1}+N_{2})$ condition by [14, Theorem
3.23].

We first show the following easy lemma.

######  Lemma 3.11.

Under Assumption 3.9, we have the following properties:

  * $(i)$

For any $f\in\operatorname{LIP}(M)$ and $x=(x_{1},x_{2})\in M$, we have that

$\operatorname{Lip}_{M}(f)(x)\geq\left(\operatorname{Lip}_{X_{1}}(f(\cdot,x_{2}))\right)(x_{1})$

and that

$\operatorname{Lip}_{M}(f)(x)\geq\left(\operatorname{Lip}_{X_{2}}(f(x_{1},\cdot))\right)(x_{2}).$

  * $(ii)$

For any $f\in\operatorname{LIP}(X_{i})$ $(i=1,2)$, we have that

$(\operatorname{Lip}_{X_{i}}f)\circ p_{i}=\operatorname{Lip}_{M}(f\circ
p_{i}).$

  * $(iii)$

For each $i=1,2$, the map $p_{i}^{\ast}\colon L^{2}(X_{i},\mathfrak{m}_{i})\to
L^{2}(M,\mathfrak{m}),\,f\mapsto f\circ p_{i}$ induces an isometric immersion
$p_{i}^{\ast}\colon W^{1,2}(X_{i})\to W^{1,2}(M)$, and we have that

$|\nabla(f\circ p_{i})|=|\nabla f|\circ p_{i}$

$\mathfrak{m}$-a.e. in $M$ for any $f\in W^{1,2}(X_{i})$ $(i=1,2)$.

  * $(iv)$

For any $f\in W^{1,2}(X_{1})$ and $h\in\operatorname{LIP}(M)$, we have that

$\langle\nabla(f\circ p_{1}),\nabla h\rangle(x)=\langle\nabla
f,\nabla(h(\cdot,x_{2}))\rangle(x_{1})$

for $\mathfrak{m}$-a.e. $x=(x_{1},x_{2})\in M$. The similar result holds for
the element of $W^{1,2}(X_{2})$.

  * $(v)$

For any $f_{i}\in W^{1,2}(X_{i})$ $(i=1,2)$, we have that

$\langle\nabla(f_{1}\circ p_{1}),\nabla(f_{2}\circ p_{2})\rangle=0$

$\mathfrak{m}$-a.e. in $M$.

###### Proof.

We get $(i)$ and $(ii)$ straightforward by the definition.

We show $(iii)$ for $i=1$. Take arbitrary $f\in W^{1,2}(X_{1})$. For any
sequence $\\{f_{n}\\}\subset\operatorname{LIP}(X_{1})$ with
$\lim\limits_{n\to\infty}\|f_{n}-f\|_{L^{2}}=0$, we have
$\lim\limits_{n\to\infty}\|f_{n}\circ p_{1}-f\circ p_{1}\|_{L^{2}}=0$, and so

$\operatorname{Ch}_{M}(f\circ
p_{1})\leq\frac{1}{2}\liminf_{n\to\infty}\int_{M}(\operatorname{Lip}_{M}(f_{n}\circ
p_{1}))^{2}\,{\rm
d}\mathfrak{m}=\frac{1}{2}\liminf_{n\to\infty}\int_{X_{1}}(\operatorname{Lip}_{X_{1}}f_{n})^{2}\,{\rm
d}\mathfrak{m}_{1}$

by $(ii)$. This implies $\operatorname{Ch}_{M}(f\circ
p_{1})\leq\operatorname{Ch}_{X_{1}}(f)$ and $f\circ p_{1}\in W^{1,2}(M)$.

We next show $\operatorname{Ch}_{M}(f\circ
p_{1})\geq\operatorname{Ch}_{X_{1}}(f)$. Take any sequence
$\\{f_{n}\\}\subset\operatorname{LIP}(M)$ with $\epsilon_{n}:=\|f_{n}-f\circ
p_{1}\|_{L^{2}}^{2}\to 0$ as $n\to\infty$. We can assume $\epsilon_{n}<1$ for
each $n$. We have that

$\displaystyle\epsilon_{n}=\int_{X_{2}}\int_{X_{1}}(f_{n}(x_{1},x_{2})-f(x_{1}))^{2}\,{\rm
d}\mathfrak{m}_{1}(x_{1})\,{\rm d}\mathfrak{m}_{2}(x_{2}),$
$\displaystyle\frac{1}{2}\int_{M}(\operatorname{Lip}_{M}f_{n})^{2}\,{\rm
d}\mathfrak{m}\geq\frac{1}{2}\int_{X_{2}}\int_{X_{1}}(\operatorname{Lip}_{X_{1}}f_{n}(\cdot,x_{2}))^{2}(x_{1})\,{\rm
d}\mathfrak{m}_{1}(x_{1})\,{\rm d}\mathfrak{m}_{2}(x_{2})$

by $(i)$, and so

$\displaystyle\mathfrak{m}_{2}\bigg{(}\bigg{\\{}x_{2}\in
X_{2}\colon\int_{X_{1}}(f_{n}(\cdot,x_{2})-f)^{2}\,{\rm
d}\mathfrak{m}_{1}>2\epsilon_{n}^{1/2}\bigg{\\}}\bigg{)}\leq\frac{1}{2}\epsilon_{n}^{1/2},$
$\displaystyle\mathfrak{m}_{2}\bigg{(}\bigg{\\{}x_{2}\in
X_{2}\colon\int_{X_{1}}(\operatorname{Lip}_{X_{1}}(f_{n}(\cdot,x_{2})))^{2}\,{\rm
d}\mathfrak{m}_{1}>(1+\epsilon_{n}^{1/2})\int_{M}(\operatorname{Lip}_{M}f_{n})^{2}\,{\rm
d}\mathfrak{m}\bigg{\\}}\bigg{)}\leq\frac{1}{1+\epsilon^{1/2}}.$

Since we have

$\frac{1}{2}\epsilon_{n}^{1/2}+\frac{1}{1+\epsilon_{n}^{1/2}}<1,$

we can take a sequence $\\{x_{2}(n)\\}\subset X_{2}$ such that

$\displaystyle\int_{X_{1}}(f_{n}(\cdot,x_{2}(n))-f)^{2}\,{\rm
d}\mathfrak{m}_{1}\leq 2\epsilon_{n}^{1/2},$
$\displaystyle\int_{X_{1}}(\operatorname{Lip}_{X_{1}}(f_{n}(\cdot,x_{2}(n))))^{2}\,{\rm
d}\mathfrak{m}_{1}\leq(1+\epsilon_{n}^{1/2})\int_{M}(\operatorname{Lip}_{M}f_{n})^{2}\,{\rm
d}\mathfrak{m}$

for each $n$. Put $g_{n}:=f_{n}(\cdot,x_{2}(n))\in\operatorname{LIP}(X_{1})$.
Then, we have $\|g_{n}-f\|_{L^{2}}\to 0$ as $n\to\infty$ and

$\operatorname{Ch}_{X_{1}}(f)\leq\frac{1}{2}\liminf_{n\to\infty}\int_{X_{1}}(\operatorname{Lip}_{X_{1}}g_{n})^{2}\,{\rm
d}\mathfrak{m}_{1}\leq\frac{1}{2}\liminf_{n\to\infty}\int_{M}(\operatorname{Lip}_{M}f_{n})^{2}\,{\rm
d}\mathfrak{m}.$

Thus, we get $\operatorname{Ch}_{M}(f\circ
p_{1})\\!\geq\\!\operatorname{Ch}_{X_{1}}(f)$, and so
$\operatorname{Ch}_{M}(f\circ p_{1})\\!=\\!\operatorname{Ch}_{X_{1}}(f)$.
Therefore, $p_{1}^{\ast}\colon W^{1,2}(X_{1})\\!\to W^{1,2}(M)$ is isometric

Let us show that $|\nabla(f\circ p_{1})|=|\nabla f|\circ p_{1}$
$\mathfrak{m}$-a.e. in $M$. Take $f_{n}\in\operatorname{LIP}(X_{1})$ such that
$f_{n}\to f$ and $\operatorname{Lip}_{X_{1}}f_{n}\to|\nabla f|$ in $L^{2}$.
Then, we have $f_{n}\circ p_{1}\to f\circ p_{1}$,
$\operatorname{Lip}_{M}(f_{n}\circ
p_{1})=(\operatorname{Lip}_{X_{1}}f_{n})\circ p_{1}\to|\nabla f|\circ p_{1}$
in $L^{2}$ and

$\operatorname{Ch}_{M}(f\circ
p_{1})=\operatorname{Ch}_{X_{1}}(f)=\frac{1}{2}\int_{M}(|\nabla f|\circ
p_{1})^{2}\,{\rm d}\mathfrak{m}.$

This implies $|\nabla(f\circ p_{1})|=|\nabla f|\circ p_{1}$
$\mathfrak{m}$-a.e. in $M$.

Let us prove $(iv)$. We first consider the case
$f\in\operatorname{Lip}(X_{1})$. Then, we have

$\displaystyle\begin{split}\langle\nabla(f\circ p_{1}),\nabla
h\rangle(x_{1},x_{2})&=\lim_{\epsilon\to
0}\frac{1}{2\epsilon}\left(\operatorname{Lip}_{M}(f\circ p_{1}+\epsilon
h)^{2}-\operatorname{Lip}_{M}(f\circ p_{1})^{2}\right)(x_{1},x_{2})\\\
&\geq\lim_{\epsilon\to
0}\frac{1}{2\epsilon}\left(\operatorname{Lip}_{X_{1}}(f+\epsilon
h(\cdot,x_{2}))^{2}-\operatorname{Lip}_{X_{1}}(f)^{2}\right)(x_{1})\\\
&=\langle\nabla f,\nabla(h(\cdot,x_{2}))\rangle(x_{1})\end{split}$

for $\mathfrak{m}$-a.e. $(x_{1},x_{2})\in M$. By considering $-h$ instead of
$h$, we also get

$\langle\nabla(f\circ p_{1}),\nabla h\rangle(x_{1},x_{2})\leq\langle\nabla
f,\nabla(h(\cdot,x_{2}))\rangle(x_{1}),$

and so

$\langle\nabla(f\circ p_{1}),\nabla h\rangle(x_{1},x_{2})=\langle\nabla
f,\nabla(h(\cdot,x_{2}))\rangle(x_{1})$

for $\mathfrak{m}$-a.e. $(x_{1},x_{2})\in M$. For general $f\in
W^{1,2}(X_{1})$, approximating $f$ by Lipschitz functions, we get $(iv)$.

Finally we show $(v)$. We have $(v)$ for each
$f_{i}\in\operatorname{LIP}(X_{i})$ ($i=1,2$) by $(iv)$. For general $f_{i}\in
W^{1,2}(X_{i})$, approximating $f_{i}$ by Lipschitz functions, we get $(v)$. ∎

We immediately get the following corollary by Lemma 3.11$(iii)$.

######  Corollary 3.12.

Under Assumption 3.9, we have that the metric measure space
$(X_{i},d_{i},m_{i})$ is infinitesimally Hilbertian and satisfies the Sobolev-
to-Lipschitz property for each $i=1,2$.

For any $f_{i}\in L^{2}(X_{i})$, we shall denote $f_{i}\circ p_{i}\in
L^{2}(M)$ by $f_{i}$ briefly if there is no confusion.

######  Lemma 3.13.

Under Assumption 3.9, we have $f_{1}f_{2}\in W^{1,2}(M)$ and

$\nabla(f_{1}f_{2})=f_{1}\nabla f_{2}+f_{2}\nabla f_{1}\in L^{2}(TM)$

for any $f_{i}\in W^{1,2}(X_{i})$ $(i=1,2)$.

###### Proof.

Take sequences
$\\{f_{i,n}\\}_{n\in\mathbb{Z}_{>0}}\subset\operatorname{LIP}(X_{i})$
($i=1,2$) such that $f_{i,n}\to f_{i}$ and
$\operatorname{Lip}_{X_{i}}(f_{i,n})\to|\nabla f_{i,n}|$ in $L^{2}(X_{i})$ as
$n\to\infty$. We have $f_{1,n}f_{2,n}\in W^{1,2}(M)$ and
$\nabla(f_{1,n}f_{2,n})=f_{1,n}\nabla f_{2,n}+f_{2,n}\nabla f_{1,n}$ by [15,
Theorem 2.2.6]. Then, $f_{1,n}f_{2,n}\to f_{1}f_{2}$ in $L^{2}(M)$ and

$\nabla(f_{1,n}f_{2,n})=f_{1,n}\nabla f_{2,n}+f_{2,n}\nabla f_{1,n}\to
f_{1}\nabla f_{2}+f_{2}\nabla f_{1}$

in $L^{2}(TM)$. Thus, we get that $f_{1,n}f_{2,n}\to f_{1}f_{2}$ in
$W^{1,2}(M)$ and $\nabla(f_{1}f_{2})=f_{1}\nabla f_{2}+f_{2}\nabla f_{1}\in
L^{2}(TM)$. ∎

Let us consider the Laplacian on $M$.

######  Lemma 3.14.

Under Assumption 3.9, we have the following properties:

  * $(i)$

For each $i=1,2$, the map $p_{i}^{\ast}\colon L^{2}(X_{i})\to L^{2}(M)$
induces a map
$p_{i}^{\ast}\colon\mathcal{D}(\Delta_{X_{i}})\to\mathcal{D}(\Delta_{M})$, and
we have that

$(\Delta_{X_{i}}f)\circ p_{i}=\Delta_{M}(f\circ p_{i})$

for any $f\in\mathcal{D}(\Delta_{X_{i}})$ $(i=1,2)$. Thus, we use the same
notation $\Delta$ for $\Delta_{M}$ and $\Delta_{X_{i}}$ $(i=1,2)$. For any
$f\in\mathcal{D}(\Delta_{X_{i}})$, we shall denote $(\Delta_{X_{i}}f)\circ
p_{i}$ by $\Delta f$ briefly if there is no confusion.

  * $(ii)$

For any $f_{i}\in\mathcal{D}(\Delta_{X_{i}})$ $(i=1,2)$, we have that
$f_{1}f_{2}\in\mathcal{D}(\Delta_{M})$ and that

$\Delta(f_{1}f_{2})=f_{1}\Delta f_{2}+(\Delta f_{1})f_{2}.$

###### Proof.

We show $(i)$ for $i=1$. Take arbitrary $f\in\mathcal{D}(\Delta_{X_{1}})$.
Then, for any $\phi\in\operatorname{LIP}(M)$, we have

$\displaystyle\int_{M}\langle\nabla(f\circ p_{1}),\nabla\phi\rangle=$
$\displaystyle\int_{X_{2}}\int_{X_{1}}\langle\nabla
f,\nabla(\phi(\cdot,x_{2}))\rangle(x_{1})\,{\rm
d}\mathfrak{m}_{1}(x_{1})\,{\rm d}\mathfrak{m}_{2}(x_{2})$ $\displaystyle=$
$\displaystyle-\int_{X_{2}}\int_{X_{1}}\Delta_{X_{1}}f(x_{1})\phi(x_{1},x_{2})\,{\rm
d}\mathfrak{m}_{1}(x_{1})\,{\rm d}\mathfrak{m}_{2}(x_{2})$ $\displaystyle=$
$\displaystyle-\int_{M}(\Delta_{X_{1}}f)\circ p_{1}\cdot\phi\,{\rm
d}\mathfrak{m}.$

Since $\operatorname{LIP}(M)\subset W^{1,2}(M)$ is dense with respect to the
norm $\|\cdot\|_{W^{1,2}}$, we get $(i)$.

We next show $(ii)$. Take arbitrary $f_{i}\in\mathcal{D}(\Delta_{X_{i}})$
($i=1,2$). Then, for any $\phi\in\operatorname{LIP}(M)$, we have

$\displaystyle\int_{M}\langle\nabla(f_{1}f_{2}),\nabla\phi\rangle=$
$\displaystyle\int_{X_{1}}\int_{X_{2}}f_{1}(x_{1})\langle\nabla
f_{2},\nabla(\phi(x_{1},\cdot))\rangle(x_{2})\,{\rm
d}\mathfrak{m}_{1}(x_{1})\,{\rm d}\mathfrak{m}_{2}(x_{2})$
$\displaystyle{}+\int_{X_{1}}\int_{X_{2}}f_{2}(x_{2})\langle\nabla
f_{1},\nabla(\phi(\cdot,x_{2}))\rangle(x_{1})\,{\rm
d}\mathfrak{m}_{1}(x_{1})\,{\rm d}\mathfrak{m}_{2}(x_{2})$ $\displaystyle=$
$\displaystyle-\int_{M}(f_{1}\Delta f_{2}+(\Delta f_{1})f_{2})\phi\,{\rm
d}\mathfrak{m}.$

Since $\operatorname{LIP}(M)\subset W^{1,2}(M)$ is dense, we get $(ii)$. ∎

Our goal is to show $(X_{2},d_{2},\mathfrak{m}_{2})$ satisfies the ${\rm
RCD}^{\ast}(K,N-n)$ condition under the assumption of Proposition 3.10.
However, we can show the following weaker assertion under Assumption 3.9.

######  Corollary 3.15.

Under Assumption 3.9, we have that the metric measure space
$(X_{i},d_{i},\mathfrak{m}_{i})$ satisfies the ${\rm RCD}^{\ast}(K,N)$
condition for each $i=1,2$.

###### Proof.

We only need to show that $(X_{i},d_{i},\mathfrak{m}_{i})$ satisfies the ${\rm
BE}(K,N)$ condition by Corollary 3.12. For any
$u_{i}\in\mathcal{D}(\Delta_{X_{i}})$ with $\Delta u_{i}\in W^{1,2}(X_{i})$
and $\phi_{i}\in\mathcal{D}(\Delta_{X_{i}})\cap L^{\infty}(X_{i})$ with
$\phi_{i}\geq 0$ and $\Delta_{X_{i}}\phi_{i}\in L^{\infty}(X_{i})$, applying
the ${\rm BE}(K,N)$ condition for $(M,d,\mathfrak{m})$ to $u_{i}\circ
p_{i},\phi_{i}\circ p_{i}\in\mathcal{D}(\Delta_{M})$, we get the ${\rm
BE}(K,N)$ condition for $(X_{i},d_{i},\mathfrak{m}_{i})$. ∎

The following proposition is crucial to show Proposition 3.10. We show the
${\rm BE}(K,N-n)$ condition with an error term.

######  Proposition 3.16.

In addition to Assumption 3.9, we assume that $n$ is an integer with $N-n\geq
1$ and that $(X_{1},d_{1},\mathfrak{m}_{1})$ is an $n$-dimensional closed
Riemannian manifold with the Riemannian distance and
$\mathfrak{m}_{1}=\underline{\mathcal{H}}^{n}$. Then, for all
$u\in\mathcal{D}(\Delta_{X_{2}})$ with $\Delta u\in W^{1,2}(X_{2})$ and all
$\phi\in\mathcal{D}(\Delta_{X_{2}})\cap L^{\infty}(X_{2})$ with $\phi\geq 0$
and $\Delta\phi\in L^{\infty}(X_{2})$, we have

$\displaystyle\frac{1}{2}\int_{X_{2}}\Delta\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2}$
$\displaystyle\qquad{}\geq\int_{X_{2}}\phi\bigg{(}\langle\nabla\Delta u,\nabla
u\rangle+K|\nabla u|^{2}+\frac{(\Delta u)^{2}}{N-n}-\frac{2n}{N(N-n)}(\Delta
u-2(N-n)u)^{2}\bigg{)}{\rm d}\mathfrak{m}_{2}.$

###### Proof.

Take $\psi\in C^{\infty}(\mathbb{R})$ such that

$\psi(t)=\begin{cases}1,&|t|\leq\dfrac{1}{2},\\\ 0,&|t|\geq 1\end{cases}$

and $\psi\geq 0$. Fix $p\in X_{1}$. Take sufficiently small $\epsilon>0$ so
that we can take $\psi_{\epsilon}$ and $f_{\epsilon}$ below as smooth
functions. Define $\psi_{\epsilon}\in C^{\infty}(X_{1})$ by

$\psi_{\epsilon}(x_{1}):=\psi\left(\frac{d(p,x_{1})}{\epsilon}\right)$

for each $x_{1}\in X_{1}$, and take $f_{\epsilon}\in C^{\infty}(X_{1})$ such
that

$f_{\epsilon}(x_{1})=\begin{cases}1+d_{1}(p,x_{1})^{2},&d(p,x_{1})\leq\epsilon,\\\
0,&d(p,x_{1})\geq 2\epsilon.\end{cases}$

Then, there exists a constant $C>0$ such that

$\displaystyle\begin{array}[]{l}\left|\operatorname{Hess}f_{\epsilon}-\frac{\Delta
f_{\epsilon}}{n}g_{X_{1}}\right|(x_{1})\leq Cd(p,x_{1}),\\\\[4.30554pt]
|\Delta f_{\epsilon}-2n|(x_{1})\leq Cd(p,x_{1}),\\\ |\nabla
f_{\epsilon}|(x_{1})\leq Cd(p,x_{1}),\\\\[4.30554pt]
|f_{\epsilon}-1|(x_{1})\leq Cd(p,x_{1})\end{array}$ (3.5)

for all $x_{1}\in B_{\epsilon}(p)$. Note that we can take such a constant
independently of $\epsilon$.

######  Claim 3.17.

There exists a constant $C>0$ such that for all sufficiently small
$\epsilon>0$, all $u\in\mathcal{D}(\Delta_{X_{2}})$ with $\Delta u\in
W^{1,2}(X_{2})$ and all $\phi\in\mathcal{D}(\Delta_{X_{2}})\cap
L^{\infty}(X_{2})$ with $\phi\geq 0$ and $\Delta\phi\in L^{\infty}(X_{2})$, we
have

$\displaystyle\frac{1}{2}\int_{X_{2}}\Delta\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2}\geq$
$\displaystyle\int_{X_{2}}\phi\bigg{(}\langle\nabla\Delta u,\nabla
u\rangle+K|\nabla u|^{2}+\frac{(\Delta u)^{2}}{N-n}\bigg{)}{\rm
d}\mathfrak{m}_{2}$
$\displaystyle{}-C\epsilon^{2}\int_{X_{2}}\phi(u^{2}+|\nabla u|^{2}+(\Delta
u)^{2})\,{\rm d}\mathfrak{m}_{2}$
$\displaystyle{}-\frac{2n}{N(N-n)}\int_{X_{2}}\phi(\Delta u-2(N-n)u)^{2}{\rm
d}\mathfrak{m}_{2}.$

###### Proof.

We have that $f_{\epsilon}u\in\mathcal{D}(\Delta_{M})$ with
$\Delta(f_{\epsilon}u)=f_{\epsilon}\Delta u+(\Delta f_{\epsilon})u\in
W^{1,2}(M)$ and that $\psi_{\epsilon}\phi\in\mathcal{D}(\Delta_{M})\cap
L^{\infty}(M)$ with $\psi_{\epsilon}\phi\geq 0$ and
$\Delta(\psi_{\epsilon}\phi)=\psi_{\epsilon}\Delta\phi+(\Delta\psi_{\epsilon})\phi\in
L^{\infty}(M)$ by Lemma 3.13 and Lemma 3.14. Thus, we can apply the ${\rm
BE}(K,N)$ condition to the pair $(f_{\epsilon}u,\psi_{\epsilon}\phi)$ and get

$\displaystyle\frac{1}{2}\int_{M}\Delta(\psi_{\epsilon}\phi)|\nabla(f_{\epsilon}u)|^{2}\,{\rm
d}\mathfrak{m}\geq\int_{M}(\psi_{\epsilon}\phi)\langle\nabla\Delta(f_{\epsilon}u),\nabla(f_{\epsilon}u)\rangle\,{\rm
d}\mathfrak{m}$
$\displaystyle\hphantom{\frac{1}{2}\int_{M}\Delta(\psi_{\epsilon}\phi)|\nabla(f_{\epsilon}u)|^{2}\,{\rm
d}\mathfrak{m}\geq}+K\int_{M}\psi_{\epsilon}\phi|\nabla(f_{\epsilon}u)|^{2}\,{\rm
d}\mathfrak{m}+\frac{1}{N}\int_{M}\psi_{\epsilon}\phi(\Delta(f_{\epsilon}u))^{2}\,{\rm
d}\mathfrak{m}.$ (3.6)

We calculate each terms.

We have

$\displaystyle\frac{1}{2}\int_{M}\Delta(\psi_{\epsilon}\phi)|\nabla(f_{\epsilon}u)|^{2}\,{\rm
d}\mathfrak{m}$
$\displaystyle\qquad=\int_{X_{1}}\psi_{\epsilon}\big{(}\langle\nabla\Delta
f_{\epsilon}\nabla f_{\epsilon}\rangle+\operatorname{Ric}(\nabla
f_{\epsilon},\nabla
f_{\epsilon})+|\operatorname{Hess}f_{\epsilon}|^{2}\big{)}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi u^{2}\,{\rm d}\mathfrak{m}_{2}$
$\displaystyle\qquad\quad{}+\int_{X_{1}}\\!\\!\psi_{\epsilon}|\nabla
f_{\epsilon}|^{2}\,{\rm d}\mathfrak{m}_{1}\int_{X_{2}}\\!\\!\phi u\Delta
u\,{\rm
d}\mathfrak{m}_{2}+\int_{X_{1}}\\!\\!\psi_{\epsilon}(f_{\epsilon}\Delta
f_{\epsilon}+2|\nabla f_{\epsilon}|^{2})\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\\!\\!\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2}$
$\displaystyle\qquad\quad{}+\frac{1}{2}\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}^{2}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\Delta\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2}.$ (3.7)

Here, we used the Bochner formula

$\frac{1}{2}\Delta|\nabla f_{\epsilon}|^{2}=\langle\nabla\Delta
f_{\epsilon}\nabla f_{\epsilon}\rangle+\operatorname{Ric}(\nabla
f_{\epsilon},\nabla f_{\epsilon})+|\operatorname{Hess}f_{\epsilon}|^{2}$

and the equation

$\int_{X_{2}}(\Delta\phi)u^{2}\,{\rm d}\mathfrak{m}_{2}=2\int_{X_{2}}\phi
u\Delta u\,{\rm d}\mathfrak{m}_{2}+2\int_{X_{2}}\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2},$

which can be justified by approximating $u$ by $P_{t}u$.

We have

$\displaystyle\int_{M}\psi_{\epsilon}\phi\langle\nabla\Delta(f_{\epsilon}u),\nabla(f_{\epsilon}u)\rangle\,{\rm
d}\mathfrak{m}$
$\displaystyle\qquad=\int_{X_{1}}\psi_{\epsilon}\langle\nabla\Delta
f_{\epsilon},\nabla f_{\epsilon}\rangle\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi u^{2}\,{\rm
d}\mathfrak{m}_{2}+\int_{X_{1}}\psi_{\epsilon}|\nabla f_{\epsilon}|^{2}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi u\Delta u\,{\rm d}\mathfrak{m}_{2}$
$\displaystyle\qquad\quad{}+\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}\Delta
f_{\epsilon}\,{\rm d}\mathfrak{m}_{1}\int_{X_{2}}\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2}+\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}^{2}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi\langle\nabla\Delta u,\nabla u\rangle\,{\rm
d}\mathfrak{m}_{2}.$ (3.8)

We have

$\displaystyle K\int_{M}\psi_{\epsilon}\phi|\nabla(f_{\epsilon}u)|^{2}\,{\rm
d}\mathfrak{m}$ $\displaystyle\qquad{}=K\int_{X_{1}}\psi_{\epsilon}|\nabla
f_{\epsilon}|^{2}\,{\rm d}\mathfrak{m}_{1}\int_{X_{2}}\phi u^{2}\,{\rm
d}\mathfrak{m}_{2}+K\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}^{2}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi|\nabla u|^{2}\,{\rm d}\mathfrak{m}_{2}.$
(3.9)

We have

$\displaystyle\frac{1}{N}\int_{M}\psi_{\epsilon}\phi(\Delta(f_{\epsilon}u))^{2}\,{\rm
d}\mathfrak{m}$
$\displaystyle\qquad{}=\frac{1}{N}\int_{X_{1}}\psi_{\epsilon}(\Delta
f_{\epsilon})^{2}\,{\rm d}\mathfrak{m}_{1}\int_{X_{2}}\phi u^{2}\,{\rm
d}\mathfrak{m}_{2}+\frac{2}{N}\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}\Delta
f_{\epsilon}\,{\rm d}\mathfrak{m}_{1}\int_{X_{2}}\phi u\Delta u\,{\rm
d}\mathfrak{m}_{2}$
$\displaystyle\qquad\quad{}+\frac{1}{N}\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}^{2}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi(\Delta u)^{2}\,{\rm d}\mathfrak{m}_{2}.$
(3.10)

Take $\widetilde{K}>0$ such that
$\operatorname{Ric}_{X_{1}}\leq\widetilde{K}g_{X_{1}}$. Then, we get

$\displaystyle\frac{1}{2}\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}^{2}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\Delta\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2}$
$\displaystyle\qquad{}\geq-\int_{X_{1}}\psi_{\epsilon}\bigg{(}(\widetilde{K}-K)|\nabla
f_{\epsilon}|^{2}+\bigg{|}\operatorname{Hess}f_{\epsilon}-\frac{\Delta
f_{\epsilon}}{n}g_{X_{1}}\bigg{|}^{2}\bigg{)}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi u^{2}\,{\rm d}\mathfrak{m}_{2}$
$\displaystyle\qquad\quad{}-2\int_{X_{1}}\psi_{\epsilon}|\nabla
f_{\epsilon}|^{2}\,{\rm d}\mathfrak{m}_{1}\int_{X_{2}}\phi|\nabla u|^{2}\,{\rm
d}\mathfrak{m}_{2}$
$\displaystyle\qquad\quad{}-\frac{1}{Nn(N-n)}\int_{M}\psi_{\epsilon}\phi((N-n)(\Delta
f_{\epsilon})u-nf_{\epsilon}\Delta u)^{2}\,{\rm d}\mathfrak{m}$
$\displaystyle\qquad\quad{}+\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}^{2}\,{\rm
d}\mathfrak{m}_{1}\int_{X_{2}}\phi\bigg{(}\langle\nabla\Delta u,\nabla
u\rangle+K|\nabla u|^{2}+\frac{1}{N-n}(\Delta u)^{2}\bigg{)}\,{\rm
d}\mathfrak{m}_{2}$

by (3.6)–(3.10). Since we have

$0<\int_{X_{1}}\psi_{\epsilon}\,{\rm
d}\mathfrak{m}_{1}\leq\int_{X_{1}}\psi_{\epsilon}f_{\epsilon}^{2}\,{\rm
d}\mathfrak{m}_{1}\leq\big{(}1+\epsilon^{2}\big{)}^{2}\int_{X_{1}}\psi_{\epsilon}\,{\rm
d}\mathfrak{m}_{1}$

and

$\displaystyle\big{\|}\sqrt{\psi_{\epsilon}\phi}((N-n)(\Delta
f_{\epsilon})u-nf_{\epsilon}\Delta u-2n(N-n)u+n\Delta u)\big{\|}_{L^{2}}$
$\displaystyle\qquad{}\leq
C\epsilon\big{(}\big{\|}\sqrt{\phi}u\big{\|}_{L^{2}}+\big{\|}\sqrt{\phi}\Delta
u\big{\|}_{L^{2}}\big{)}\bigg{(}\int_{X_{1}}\psi_{\epsilon}\,{\rm
d}\mathfrak{m}_{1}\bigg{)}^{1/2},$

we get the claim by (3.5). ∎

Letting $\epsilon\to 0$ in Claim 3.17, we get the proposition. ∎

Let us complete the proof of Proposition 3.10. Since we have already showed
Corollary 3.12, we only need to check the ${\rm BL}(K,N-n)$ condition for
$(X_{2},d_{2},\mathfrak{m}_{2})$. The proof of the following proposition has
been inspired by the proof of [24, Theorem 1.2].

######  Proposition 3.18.

In addition to Assumption 3.9, we assume that $n$ is an integer with $N-n\geq
1$ and that $(X_{1},d_{1},\mathfrak{m}_{1})$ is an $n$-dimensional closed
Riemannian manifold with the Riemannian distance and
$\mathfrak{m}_{1}=\underline{\mathcal{H}}^{n}$. Then, the metric measure space
$(X_{2},d_{2},\mathfrak{m}_{2})$ satisfies the ${\rm BL}(K,N-n)$ condition.

###### Proof.

Similarly to the proof of the assertion that the ${\rm BE}(K,N)$ condition
implies the ${\rm BL}(K,N)$ condition [14, Proposition 4.9], we have the
following claim:

######  Claim 3.19.

For any $u\in\mathcal{D}(\Delta_{X_{2}})$ and $t>0$, we have

$|\nabla P_{t}u|^{2}+\frac{1-{\rm e}^{-2Kt}}{K(N-n)}\bigg{(}(\Delta
P_{t}u)^{2}-\frac{2n}{N}P_{t}\big{(}(\Delta
u-2(N-n)u)^{2}\big{)}\bigg{)}\leq{\rm e}^{-2Kt}P_{t}\big{(}|\nabla
u|^{2}\big{)}$

$\mathfrak{m}_{2}$-a.e. in $X_{2}$.

###### Proof.

Take arbitrary $\phi\in L^{\infty}(X_{2})$ with $\phi\geq 0$. Define
$h\colon[0,t]\to\mathbb{R}$ by

$h(s):={\rm e}^{-2Ks}\int_{X_{2}}P_{s}\phi|\nabla P_{t-s}u|^{2}\,{\rm
d}\mathfrak{m}_{2}.$

Then, for each $0<s<t$, we have

$\displaystyle\frac{\partial}{\partial s}h(s)=-2K{\rm
e}^{-2Ks}\int_{X_{2}}P_{s}\phi|\nabla P_{t-s}u|^{2}\,{\rm d}\mathfrak{m}_{2}$
$\displaystyle\hphantom{\frac{\partial}{\partial s}h(s)=}{}+{\rm
e}^{-2Ks}\int_{X_{2}}\Delta P_{s}\phi|\nabla P_{t-s}u|^{2}\,{\rm
d}\mathfrak{m}_{2}-2{\rm e}^{-2Ks}\int_{X_{2}}P_{s}\phi\langle\nabla\Delta
P_{t-s}u,\nabla P_{t-s}u\rangle\,{\rm d}\mathfrak{m}_{2}$
$\displaystyle\hphantom{\frac{\partial}{\partial s}h(s)}{}\geq 2\frac{{\rm
e}^{-2Ks}}{N-n}\bigg{(}\int_{X_{2}}\\!\\!P_{s}\phi(\Delta P_{t-s}u)^{2}\,{\rm
d}\mathfrak{m}_{2}-\frac{2n}{N}\\!\int_{X_{2}}\\!\\!P_{s}\phi(\Delta
P_{t-s}u-2(N-n)P_{t-s}u)^{2}{\rm d}\mathfrak{m}_{2}\\!\bigg{)}$
$\displaystyle\hphantom{\frac{\partial}{\partial s}h(s)}{}\geq 2\frac{{\rm
e}^{-2Ks}}{N-n}\bigg{(}\int_{X_{2}}\phi(\Delta P_{t}u)^{2}\,{\rm
d}\mathfrak{m}_{2}-\frac{2n}{N}\int_{X_{2}}\phi P_{t}\big{(}(\Delta
u-2(N-n)u)^{2}\big{)}\,{\rm d}\mathfrak{m}_{2}\bigg{)}.$

Here, we used

$\displaystyle\frac{\partial}{\partial s}P_{s}\phi=\Delta P_{s}\phi,$
$\displaystyle\frac{\partial}{\partial s}P_{t-s}u=-\Delta P_{t-s}u$

in $W^{1,2}$, $|\nabla P_{t-s}u|^{2}\leq{\rm e}^{-2K(t-s)}P_{t-s}(|\nabla
u|^{2})$ (by the ${\rm BL}(K,N)$ condition), Proposition 3.16 and the Jensen
inequality

$\displaystyle(\Delta P_{t}u)^{2}\leq P_{s}\big{(}(\Delta
P_{t-s}u)^{2}\big{)},$ $\displaystyle(\Delta P_{t-s}u-2(N-n)P_{t-s}u)^{2}\leq
P_{t-s}\big{(}(\Delta u-2(N-n)u)^{2}\big{)}$

$\mathfrak{m}_{2}$-a.e. in $X_{2}$. Combining this and

$\displaystyle\lim_{s\to 0}h(s)=h(0)=\int_{X_{2}}\phi|\nabla P_{t}u|^{2}\,{\rm
d}\mathfrak{m}_{2},$ $\displaystyle\lim_{s\to t}h(s)=h(t)={\rm
e}^{-2Kt}\int_{X_{2}}\phi P_{t}|\nabla u|^{2}\,{\rm d}\mathfrak{m}_{2},$

we get

$\displaystyle\int_{X_{2}}\phi\left({\rm e}^{-2Kt}P_{t}|\nabla u|^{2}-|\nabla
P_{t}u|^{2}\right)\,{\rm d}\mathfrak{m}_{2}$
$\displaystyle\qquad{}\geq\frac{1-{\rm
e}^{-2Kt}}{K(N-n)}\left(\int_{X_{2}}\phi(\Delta P_{t}u)^{2}\,{\rm
d}\mathfrak{m}_{2}-\frac{2n}{N}\int_{X_{2}}\phi P_{t}\left((\Delta
u-2(N-n)u)^{2}\right)\,{\rm d}\mathfrak{m}_{2}\right).$

This implies the claim. ∎

Let us show that $(X_{2},d_{2},\mathfrak{m}_{2})$ satisfies the ${\rm
BL}(K,N-n)$ condition. Take $u\in W^{1,2}(X_{2})$ and fix $s>0$. Define

$v:=\Delta P_{s}u-2(N-n)P_{s}u=P_{s/2}\Delta
P_{s/2}u-2(N-n)P_{s}u\in\operatorname{TestF}(X_{2})$

and $v_{x}:=v-v(x)$ for each $x\in X_{2}$. Then, the functions $v_{x}\colon
X_{2}\to\mathbb{R}$ and

$[0,\infty)\times X_{2}\to\mathbb{R},\qquad(t,y)\mapsto P_{t}(v_{x}^{2})(y)$

are continuous. Thus, for fixed $\epsilon>0$ and any $x\in X_{2}$, there
exists $\delta_{x},\tau_{x}>0$ such that we have

$|P_{t}(v_{x}^{2})(y)|<\epsilon$

for any $y\in B_{\delta_{x}}(x)$ and $t\in(0,\tau_{x})$. Since $X_{2}$ is
compact, there exist points $x_{1},\ldots,x_{k}\in X_{2}$
($k\in\mathbb{Z}_{>0}$) such that

$X_{2}=\bigcup_{i}B_{\delta_{x_{i}}}(x_{i}).$

Put $\tau:=\min{\tau_{x_{i}}}$. Define

$\tilde{v}_{i}:=P_{s}u-(P_{s}u)(x_{i})+\frac{1}{2(N-n)}\Delta(P_{s}u)(x_{i}).$

Then, we have $\tilde{v}_{i}\in\mathcal{D}(\Delta_{X_{2}})$ and

$\Delta\tilde{v}_{i}-2(N-n)\tilde{v}_{i}=\Delta
P_{s}u-2(N-n)P_{s}u+2(N-n)(P_{s}u)(x_{i})-\Delta(P_{s}u)(x_{i})=v_{x_{i}}.$

Applying Claim 3.19 to $\tilde{v}_{i}$, for each $i$ and $t\in(0,\tau)$, we
get

$\displaystyle{\rm e}^{-2Kt}P_{t}\big{(}|\nabla P_{s}u|^{2}\big{)}$
$\displaystyle\geq|\nabla P_{s+t}u|^{2}+\frac{1-{\rm
e}^{-2Kt}}{K(N-n)}\bigg{(}(\Delta
P_{s+t}u)^{2}-\frac{2n}{N}P_{t}\big{(}v_{x_{i}}^{2}\big{)}\bigg{)}$
$\displaystyle\geq|\nabla P_{s+t}u|^{2}+\frac{1-{\rm
e}^{-2Kt}}{K(N-n)}\bigg{(}(\Delta P_{s+t}u)^{2}-\frac{2n}{N}\epsilon\bigg{)}$

$\mathfrak{m}_{2}$-a.e. in $B_{\delta_{x_{i}}}(x_{i})$. Thus, for each $i$ and
$t\in(0,\tau)$, we get

${\rm e}^{-2Kt}P_{t}\big{(}|\nabla P_{s}u|^{2}\big{)}\geq|\nabla
P_{s+t}u|^{2}+\frac{1-{\rm e}^{-2Kt}}{K(N-n)}\left((\Delta
P_{s+t}u)^{2}-\frac{2n}{N}\epsilon\right)$

$\mathfrak{m}_{2}$-a.e. in $X_{2}$. Letting $\epsilon\to 0$, we get

${\rm e}^{-2Kt}P_{t}\big{(}|\nabla P_{s}u|^{2}\big{)}\geq|\nabla
P_{s+t}u|^{2}+\frac{1-{\rm e}^{-2Kt}}{K(N-n)}(\Delta P_{s+t}u)^{2}$

$\mathfrak{m}_{2}$-a.e. in $X_{2}$. Letting $s\to 0$, we get the following
inequality as the limit in $L^{1}(X_{2})$:

${\rm e}^{-2Kt}P_{t}\big{(}|\nabla u|^{2}\big{)}\geq|\nabla
P_{t}u|^{2}+\frac{1-{\rm e}^{-2Kt}}{K(N-n)}(\Delta P_{t}u)^{2}$

$\mathfrak{m}_{2}$-a.e. in $X_{2}$. This is the ${\rm BL}(K,N-n)$ condition. ∎

By Corollary 3.12 and Proposition 3.18, we get Proposition 3.10. By
Propositions 3.6 and 3.10, we get Proposition 3.2.

##  4 Proof of the Main Theorem

In this section we complete the proof of our main theorem.

######  Theorem 4.1.

For integers $n\geq 5$ and $2\leq p<n/2$ and a positive real number
$\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that the
following property holds. Let $(M,g)$ be an $n$-dimensional closed Riemannian
manifold with $\operatorname{Ric}_{g}\geq(n-p-1)g$, and assume one of the
following:

  * •

$\lambda_{1}(\Delta_{C,p})\leq\delta$ and $\lambda_{n-p+1}(g)\leq n-p+\delta$,

  * •

$\lambda_{1}(\Delta_{C,n-p})\leq\delta$ and $\lambda_{n-p}(g)\leq n-p+\delta$.

Then, we have

$\displaystyle\lambda_{n-p+2}(g)\geq\frac{p(n-p-1)}{p-1}-\epsilon.$

By the Lichnerowicz estimate for the first eigenvalue of the Laplacian acting
on functions for metric measure spaces satisfying the ${\rm
RCD}^{\ast}(n-p-1,p)$ condition [14, Theorem 4.22]:

$\lambda_{1}\geq\frac{p(n-p-1)}{p-1},$

we get Theorem 4.1 similarly to Theorem 4.2 below. Thus, we only give the
proof of Theorem 4.2.

The following theorem is the main result of this article.

######  Theorem 4.2.

For integers $n\geq 5$ and $2\leq p<n/2$ and a positive real number
$\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that the
following property holds. Let $(M,g)$ be an $n$-dimensional closed Riemannian
manifold with $\operatorname{Ric}_{g}\geq(n-p-1)g$ satisfying one of the
following:

  * •

$\lambda_{1}(\Delta_{C,p})\leq\delta$, $\lambda_{n-p+1}(g)\leq n-p+\delta$ and
$\lambda_{n+1}(g)\leq p(n-p-1)/(p-1)+\delta$,

  * •

$\lambda_{1}(\Delta_{C,n-p})\leq\delta$, $\lambda_{n-p}(g)\leq n-p+\delta$ and
$\lambda_{n+1}(g)\leq p(n-p-1)/(p-1)+\delta$.

Then, we have

$d_{{\rm GH}}\left(M,S^{n-p}(1)\times
S^{p}\left(\sqrt{\frac{p-1}{n-p-1}}\right)\right)\leq\epsilon.$

###### Proof.

We show the theorem by a contradiction. Suppose that the theorem does not
hold. Then, there exists a sequence of $n$-dimensional closed Riemannian
manifolds $\\{(M_{i},g_{i})\\}_{i=1}^{\infty}$ with
$\operatorname{Ric}_{g_{i}}\geq(n-p-1)g_{i}$ that does not converge to
$S^{n-p}(1)\times S^{p}\big{(}\sqrt{(p-1)/(n-p-1)}\big{)}$ and that satisfies
one of the following:

  * •

$\lim\limits_{i\to\infty}\lambda_{n-p+1}(g_{i})=n-p$,
$\lim\limits_{i\to\infty}\lambda_{n+1}(g_{i})=p(n-p-1)/(p-1)$ and
$\lim\limits_{i\to\infty}\lambda_{1}(\Delta_{C,p},g_{i})=0$,

  * •

$\lim\limits_{i\to\infty}\lambda_{n-p}(g_{i})=n-p$,
$\lim\limits_{i\to\infty}\lambda_{n+1}(g_{i})=p(n-p-1)/(p-1)$ and
$\lim\limits_{i\to\infty}\lambda_{1}(\Delta_{C,n-p},g_{i})=0$.

Taking a subsequence, we have that Assumption 3.1 holds by Propositions 2.18
and 2.19, the Gromov compactness theorem (see also [27, Theorem 11.1.10], [1,
Theorem 4.54]) and [12, Theorems 1.6 and 1.10]. Then, there exists a Borel
measure $\mathfrak{m}_{X}$ on X such that
$\mathfrak{m}=\underline{\mathcal{H}}^{n-p}\times\mathfrak{m}_{X}$ holds and
$(X,\mathfrak{m}_{X})$ satisfies the ${\rm RCD}^{\ast}(n-p-1,p)$ condition by
Proposition 3.2. By the spectral convergence theorem [13, Theorem 7.9] and
Theorem 4.1, we have

$\lambda_{n-p+2}\big{(}S^{n-p}(1)\times
X,\mathfrak{m}\big{)}=\cdots=\lambda_{n+1}\big{(}S^{n-p}(1)\times
X,\mathfrak{m}\big{)}=\frac{p(n-p-1)}{p-1}.$

Since the spectrum of the Laplacian on $(S^{n-p}(1)\times X,\mathfrak{m})$
coincides with

$\big{\\{}\lambda_{i}\big{(}S^{n-p}(1),\underline{\mathcal{H}}^{n-p}\big{)}+\lambda_{j}(X,\mathfrak{m}_{X})\colon
i,j\in\mathbb{Z}_{\geq 0}\big{\\}}$

and
$\lambda_{n-p+2}\big{(}S^{n-p}(1),\underline{\mathcal{H}}^{n-p}\big{)}=2(n-p+1)>p(n-p-1)/(p-1)$,
we get that

$\lambda_{1}(X,\mathfrak{m}_{X})=\cdots=\lambda_{p}(X,\mathfrak{m}_{X})=\frac{p(n-p-1)}{p-1}.$

By the Obata Rigidity theorem for metric measure spaces satisfying the ${\rm
RCD}^{\ast}$ condition [25, Theorem 1.4] with scaling, we have that
$(X,\mathfrak{m}_{X})$ is isomorphic to either
$\big{(}S^{p}(r_{n,p}),\underline{\mathcal{H}}^{p}\big{)}$ or
$\big{(}S^{p}_{+}(r_{n,p}),\underline{\mathcal{H}}^{p}\big{)}$, where
$r_{n,p}:=\sqrt{(p-1)/(n-p-1)}$ and $S^{p}_{+}(r_{n,p})$ denotes the
$p$-dimensional hemisphere with radius $r_{n,p}$. In particular,
$\big{\\{}\big{(}M_{i},g_{i},\underline{\mathcal{H}}^{n}\big{)}\big{\\}}$ is a
non-collapsing sequence. Thus, we get $(X,\mathfrak{m}_{X})$ is isomorphic to
$\big{(}S^{p}(r_{n,p}),\underline{\mathcal{H}}^{p}\big{)}$ by [12, Theorem
6.2]. This contradicts to the assumption, and so we get the theorem. ∎

### Acknowledgements

I am grateful to Professor Shouhei Honda for helpful discussions. I also thank
Professor Dario Trevisan for answering my questions about the regular
Lagrangian flow. I wish to thank the referees for careful reading of the paper
and making valuable suggestions. This work was supported by RIKEN Special
Postdoctoral Researcher Program.

## References

  * [1] Aino M., Lichnerowicz–Obata estimate, almost parallel $p$-form and almost product manifolds, arXiv:1904.06533.
  * [2] Ambrosio L., Gigli N., Savaré G., Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014), 289–391, arXiv:1106.2090.
  * [3] Ambrosio L., Gigli N., Savaré G., Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405–1490, arXiv:1109.0222.
  * [4] Ambrosio L., Gigli N., Savaré G., Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab. 43 (2015), 339–404, arXiv:1209.5786.
  * [5] Ambrosio L., Honda S., New stability results for sequences of metric measure spaces with uniform Ricci bounds from below, in Measure Theory in Non-Smooth Spaces, Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017, 1–51, arXiv:1605.07908.
  * [6] Ambrosio L., Trevisan D., Well-posedness of Lagrangian flows and continuity equations in metric measure spaces, Anal. PDE 7 (2014), 1179–1234, arXiv:1402.4788.
  * [7] Ambrosio L., Trevisan D., Lecture notes on the DiPerna–Lions theory in abstract measure spaces, Ann. Fac. Sci. Toulouse Math. 26 (2017), 729–766, arXiv:1505.05292.
  * [8] Aubry E., Pincement sur le spectre et le volume en courbure de Ricci positive, Ann. Sci. École Norm. Sup. (4) 38 (2005), 387–405, arXiv:math.DG/0505408.
  * [9] Burago D., Burago Y., Ivanov S., A course in metric geometry, Graduate Studies in Mathematics, Vol. 33, Amer. Math. Soc., Providence, RI, 2001.
  * [10] Cavalletti F., Milman E., The globalization theorem for the curvature dimension condition, arXiv:1612.07623.
  * [11] Cheeger J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428–517.
  * [12] Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406–480.
  * [13] Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), 37–74.
  * [14] Erbar M., Kuwada K., Sturm K.-T., On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015), 993–1071, arXiv:1303.4382.
  * [15] Gigli N., Nonsmooth differential geometry – an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 (2018), v+161, arXiv:1407.0809.
  * [16] Gigli N., Pasqualetto E., Lectures on nonsmooth differential geometry, SISSA Springer Series, Vol. 2, Springer, 2020.
  * [17] Gigli N., Rigoni C., Recognizing the flat torus among ${\rm RCD}^{*}(0,N)$ spaces via the study of the first cohomology group, Calc. Var. Partial Differential Equations 57 (2018), 104, 39 pages, arXiv:1705.04466.
  * [18] Grosjean J.-F., A new Lichnerowicz–Obata estimate in the presence of a parallel $p$-form, Manuscripta Math. 107 (2002), 503–520.
  * [19] Hajłasz P., Koskela P., Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1211–1215.
  * [20] Hajłasz P., Koskela P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), x+101 pages.
  * [21] Honda S., Ricci curvature and almost spherical multi-suspension, Tohoku Math. J. 61 (2009), 499–522.
  * [22] Honda S., Ricci curvature and $L^{p}$-convergence, J. Reine Angew. Math. 705 (2015), 85–154, arXiv:1212.2052.
  * [23] Jiang R., Li H., Zhang H., Heat kernel bounds on metric measure spaces and some applications, Potential Anal. 44 (2016), 601–627, arXiv:1407.5289.
  * [24] Ketterer C., Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl. 103 (2015), 1228–1275, arXiv:1311.1307.
  * [25] Ketterer C., Obata’s rigidity theorem for metric measure spaces, Anal. Geom. Metr. Spaces 3 (2015), 278–295, arXiv:1410.5210.
  * [26] Petersen P., On eigenvalue pinching in positive Ricci curvature, Invent. Math. 138 (1999), 1–21.
  * [27] Petersen P., Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, Vol. 171, Springer, Cham, 2016.
  * [28] Rajala T., Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations 44 (2012), 477–494, arXiv:1107.4842.
  * [29] Sturm K.-T., Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995), 275–312.
  * [30] Sturm K.-T., Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. 75 (1996), 273–297.
  * [31] Sturm K.-T., On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65–131.
  * [32] Sturm K.-T., On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133–177.