# Probabilistic Method to Fundamental gap problems on the sphere

Gunhee Cho  Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA 93106<EMAIL_ADDRESS>,  Guofang Wei  Department
of Mathematics
University of California, Santa Barbara
Santa Barbara, CA 93106<EMAIL_ADDRESS>and  Guang Yang  Department of
Mathematics
Purdue University, West Lafayette, IN 47907<EMAIL_ADDRESS>
###### Abstract.

We provide a probabilistic proof of the fundamental gap estimate for
Schrödinger operators in convex domains on the sphere, which extends the
probabilistic proof of F. Gong, H. Li, and D. Luo for the Euclidean case. Our
results further generalize the results achieved for the Laplacian by S. Seto,
L. Wang, and G. Wei, as well as by C. He, G. Wei, and Qi S. Zhang. The
essential ingredient in our analysis is the reflection coupling method on
Riemannian manifolds.

GC was partially supported by AMS Simons travel grant

GW was partially supported by NSF DMS grant 2104704.

##  1\. Introduction

In the context of a bounded smooth domain $\Omega$ within a Riemannian
manifold, the eigenvalues of the Laplacian or more generally, Schrödinger
operators on $\Omega$ subject to Dirichlet boundary conditions are ordered as
follows:

$\lambda_{1}<\lambda_{2}\leq\lambda_{3}\cdots\rightarrow\infty.$

The estimate of the gap between the first two eigenvalues, known as the
fundamental (or mass) gap, denoted as

$\Gamma(\Omega)=\lambda_{2}-\lambda_{1}>0,$

holds significant importance in both mathematics and physics, and has been a
subject of active study.

In a noteworthy achievement, using two point maximal principle B. Andrews and
J. Clutterbuck successfully proved the fundamental gap conjecture for convex
domains in Euclidean space [1]. Namely, for Schrödinger operators with convex
potential, we have $\Gamma(\Omega)\geq\tfrac{3\pi^{2}}{D^{2}}$ for convex
domains $\Omega\subset\mathbb{R}^{n}$, where $D$ is the diameter of the
domain. More recently, the second author, along with coauthors, extended the
same estimate to convex domains on the sphere for the Laplacian [18, 9, 6].

An essential step in proving the fundamental gap conjecture is the log-
concavity estimate of the first eigenfunction. The following definition was
firstly introduced by B. Andrews and J. Clutterbuck [1]. An even function
$\tilde{V}\in C^{1}([-D/2,D/2])$ is called a modulus of concavity for $V\in
C^{1}(\Omega)$, if for all $x,y\in\Omega,x\neq y$, one has

$\left\langle\nabla V(y),\gamma^{\prime}(y)\right\rangle-\left\langle\nabla
V(x),\gamma^{\prime}(x)\right\rangle\leq
2\tilde{V}^{\prime}\left(\frac{\rho(x,y)}{2}\right),$ (1.1)

where $\gamma$ represents the minimizing geodesic that connects $x,y$ and
$\rho$ denotes the geodesic distance function. Intuitively, this says that $V$
is “more concave” than $\tilde{V}$. If the inequality (1.1) is reversed

$\left\langle\nabla V(y),\gamma^{\prime}(y)\right\rangle-\left\langle\nabla
V(x),\gamma^{\prime}(x)\right\rangle\geq
2\tilde{V}^{\prime}\left(\frac{\rho(x,y)}{2}\right),$ (1.2)

we say $\tilde{V}$ is a modulus of convexity for $V$.

Let $\phi_{1}$ be the first positive Dirichlet eigenfunction of the
Schrödinger operator $-\Delta+V$ on $\Omega$ and $\bar{\phi}_{1}$ the first
Dirichlet eigenfunction of the one-dimensional model given in (1.3). The key
estimates in obtaining the fundamental gap lower bound is the following log-
concavity estimate of the first eigenfunction.

$\langle\nabla\log\phi_{1}(y),\gamma^{\prime}(y)\rangle-\langle\nabla\log\phi_{1}(x),\gamma^{\prime}(x)\rangle\leq
2\log\bar{\phi}_{1}^{\prime}(\rho(x,y)/2)$

for any $x\neq y\in\Omega$. Such log-concavity estimate was proved for
Schrödinger operators in convex domains $\Omega\subset\mathbb{R}^{n}$ in [1],
and for the Laplacian in convex domains $\Omega\subset\mathbb{S}^{n}$ with
diameter $D<\pi/2$ in [18] 111The Schrödinger operator case was stated in [18,
Remark 1.4] but no proof was given.. The diameter restriction was removed in
[9].

In this paper we give a probabilistic proof of this estimate. Denote by
$\mathbb{M}^{n}_{k}$ the $n$-dimensional simply connected manifolds with
constant sectional curvature $k$. Our first main result is the next

######  Theorem 1.1.

Suppose $\Omega\subset\mathbb{M}^{n}_{k}$ $(k\geq 0)$ is a bounded strictly
convex domain with diameter $D<\pi/\sqrt{k}$. Let $\phi_{1}>0$ be the first
Dirichlet eigenfunction of $-\Delta+V$ on $\Omega$ and $\bar{\phi}_{1}$ be the
first Dirichlet eigenfunction of the one-dimensional model

$-\frac{\partial^{2}}{\partial s^{2}}+(n-1)tn_{k}(s)\frac{\partial}{\partial
s}+\tilde{V}(s),\ s\in[-D/2,D/2],$ (1.3)

where $tn_{k}$ is defined in (2.7). If $\tilde{V}$ is a modulus of convexity
of $V$, then $\log\bar{\phi}_{1}$ is a modulus of concavity of $\log\phi_{1}$.

The above theorem extends the results in [18, 9] for the Laplacian operator to
Schrödinger operators. As mentioned before, our argument is based on the
reflection coupling method (see for instance, [5, 14, 10]), which has been
used in [8] to give a probabilistic proof for convex domain in the Euclidean
space. Some of our computations are similar to that of [18, 9], but the whole
proof is much simplified. Namely, the parabolic analysis in [9] is not needed
at all. Also we do not need the initial analysis or log-concavity assumption
of the first eigenfunction as in [8]. It is also worth mentioning that
coupling methods have been used to derive lower bounds for the first nonzero
Neumann eigenvalue [4, 2, 3].

With the log-concavity estimate, another ingredient for getting the
fundamental gap estimate is the following gap comparison of [18, Theorem 4.1].
We extend it to Schrödinger operators with a probabilistic proof, which
simplifies the original proof.

######  Theorem 1.2.

Let $(M^{n},g)$ be a complete Riemannian manifold with Ricci curvature lower
bound $(n-1)k,k\in\mathbb{R}$ and $\Omega\subset M^{n}$ be a convex domain
with diameter $D>0$ ($D<\pi/\sqrt{k}$ if $k>0$). Let $\lambda_{1},\lambda_{2}$
be the first two Dirichlet eigenvalues of the Schrödinger operator $-\Delta+V$
on $\Omega$ and $\phi_{1}$ be the corresponding first eigenfunction. Let
$\bar{\lambda}_{1},\bar{\lambda}_{2}$ be the first two Dirichlet eigenvalues
of the one-dimensional model (1.3) on $[-D/2,D/2]$ and $\tilde{\phi}_{1}$ be
the corresponding first eigenfunction. If $\log\tilde{\phi}_{1}$ is a modulus
of concavity of $\log\phi_{1}$, then

$\Gamma(\Omega)=\lambda_{2}-\lambda_{1}\geq\bar{\lambda}_{2}(n,k,D)-\bar{\lambda}_{1}(n,k,D).$

The above two theorems immediately give the following.

######  Corollary 1.1.

Let $\Omega$ be a strictly convex domain in $\mathbb{M}_{k}$ ($k\geq 0$) with
diameter $D<\pi/\sqrt{k}$. Let $\lambda_{i}(i=1,2)$ be the first two Dirichlet
eigenvalues of the Schrödinger operator $-\Delta+V$ on $\Omega$, and
$\tilde{V}$ is a modulus of convexity of $V$. Then

$\Gamma(\Omega)=\lambda_{2}-\lambda_{1}\geq\bar{\lambda}_{2}(n,k,D)-\bar{\lambda}_{1}(n,k,D)$

where $\bar{\lambda}_{i}(n,k,D)(i=1,2)$ are the first two Dirichlet
eigenvalues of the operator (1.3) on $[-D/2,D/2]$.

This fundamental gap estimate recovers the estimate in [1], and extends the
estimate in [18, 9] to Schrödinger operators.

The rest of this paper is structured as follows. Section 2 provides some
necessary background materials for both Riemannian geometry and stochastic
analysis. We define and study two diffusion couplings in Section 3, which will
be used in our later proofs. Section 4 is devoted to the proof of the log-
concavity of the first eigenfunction, and a proof for the gap comparison is
given in Section 5.

##  2\. Preliminaries

###  2.1. Brownian motion and diffusion in Euclidean space

In this subsection, we gather the essential materials from the classical
stochastic analysis. These materials here are well-known; we include them so
that readers without any probability background can acquire intuitions and
basic understandings. Some excellent and comprehensive presentations of these
topics can be found in [13, 17, 16, 7].

We fix a complete probability space $(A,\mathcal{F},\mathbb{P})$. Here $A$ is
the sample space, $\mathcal{F}$ is a $\sigma$-field of of subsets of $A$, and
$\mathbb{P}$ a complete probability measure on $(A,\mathcal{F})$.

######  Definition 2.1 (Real Brownian motion).

A real-valued process $\\{W_{t}\\}_{t\geq 0}$ defined on $A$ is said to be a
standard Brownian motion if

  1. (1)

$W_{0}=0$.

  2. (2)

For each $t>0$, $W_{t}$ is Gaussian with mean $0$ and variance $t$.

  3. (3)

$W_{t}$ has independent increments; that is, $W_{t}-W_{s}$ and $W_{u}-W_{v}$
are independent whenever $[s,t]\cap[u,v]=\emptyset$.

  4. (4)

$W_{t}$ is continuous as a function of $t\in[0,+\infty)$.

More generally, for any $n\geq 1$, we can define a standard Brownian motion as
a $\mathbb{R}^{n}$-valued process whose components are independent standard
real Brownian motions.

Let $\\{\mathcal{F}_{t}\\}_{t\geq 0}$ be a family of increasing sub
$\sigma$-field of $\mathcal{F}$. A process $\left\\{X_{t}\right\\}_{t\geq 0}$
is called $\mathcal{F}_{t}$-adapted if $X_{t}$ is $\mathcal{F}_{t}$ measurable
for any $t\geq 0$ (i.e. for any real numbers $a<b$, we have $\\{\omega\in
A:a<X_{t}(\omega)<b\\}\subset\mathcal{F}_{t}$).

######  Definition 2.2.

A process $\left\\{X_{t}\right\\}_{t\geq 0}$ is said to be a
$\mathcal{F}_{t}$-martingale if $X_{t}$ is $\mathcal{F}_{t}$-adapted and
integrable for every $t\geq 0$, and the conditional expectation
$\mathbb{E}\left(X_{t}\mid\mathcal{F}_{s}\right)$ is equal to $X_{s}$ almost
surely, for all $0\leq s\leq t$. In particular,
$\mathbb{E}(X_{t})=\mathbb{E}(X_{0})$.

Brownian motion is the most classical example of martingales. The associated
$\mathcal{F}_{t}$ is given by the $\sigma$-field generated by $\\{\omega\in
A:a<W_{s}(\omega)<b,\\}$, where $0\leq s\leq t$ and $a<b$ are any real
numbers. We will call $\mathcal{F}_{t}$ the $\sigma$-field generated by
$\\{W_{s}\\}_{0\leq s\leq t}$.

For any $t>0$, a Brownian motion has unbounded variation on $[0,t]$. More
precisely,

$\lim_{\left|\Delta t_{i}\right|\rightarrow
0}\sum_{i}\left|W_{t_{i+1}}-W_{t_{i}}\right|=+\infty$

almost surely, where $\\{t_{i}\\}_{i\geq 1}$ is any partition of $[0,t]$ with
mesh $\Delta t_{i}$. As a result, it is in general impossible to define
integrals against Brownian motions as Riemann-Stieltjes integrals. To properly
define stochastic integrals, we will need the so called quadratic variation,
which we describe now.

Suppose $X_{t}$ is a square-integrable martingale (i.e.
$\mathbb{E}(X^{2}_{t})<+\infty$ for all $t\geq 0$), the quadratic variation of
$X_{t}$ is defined as

$\langle X\rangle_{t}=\lim_{\left|\Delta t_{i}\right|\rightarrow
0}\sum_{i}\left|X_{t_{i+1}}-X_{t_{i}}\right|^{2},$

where the limit is taken in the probability measure $\mathbb{P}$. If two
square integrable martingales $X_{t},Y_{t}$ are adapted to the same filtration
$\mathcal{F}_{t}$, then their linear combinations are also square integrable
martingales. We define their cross variation as

$\langle X,Y\rangle_{t}=\frac{1}{4}\left(\langle X+Y\rangle_{t}-\langle
X-Y\rangle_{t}\right).$

The process $\langle X\rangle_{t}$ is always of bounded variation. In
particular, we can find a locally bounded function $h(X)$ such that $d\langle
X\rangle_{t}=h(X)_{t}dt$. We can give another characterization of standard
Brownian motions through quadratic variation.

######  Proposition 2.3 (Levy’s characterization).

If $X_{t}$ is a continuous square integrable martingale with $\langle
X\rangle_{t}=t$ such that $\mathbb{P}(X_{0}=0)=1$, then $X_{t}$ is a standard
real Brownian motion.

######  Proposition 2.4.

Let $Z_{t}$ be a $\mathcal{F}_{t}$-adapted process and $X_{t}$ be a square
integrable $\mathcal{F}_{t}$-martingale such that

$\int_{0}^{T}Z^{2}_{s}d\langle X\rangle_{s}<+\infty.$

Then, along any partitions of $[0,T]$, the limit

$\lim_{\left|\Delta t_{i}\right|\rightarrow
0}\sum_{i}Z_{t_{i}}(X_{t_{i+1}}-X_{t_{i}})$

exists in $L^{2}(A)$. This limit will be denoted as $\int_{0}^{t}Z_{s}dX_{s}.$

We can now define stochastic integration.

######  Definition 2.5 (Itô’s integral).

The limit in Proposition 2.4 is called the Itô’s integral of $Z_{t}$ against
$X_{t}$, and is denoted by $I_{t}(Z)=\int^{t}_{0}Z_{s}dX_{s}$. The quadratic
variation and cross variation of Itô’s integral are given as

$\langle I(Z)\rangle_{t}=\int_{0}^{t}Z^{2}_{s}d\langle X\rangle_{s},\ \ \ \
\langle I(Z),I(W)\rangle_{t}=\int^{t}_{0}Z_{s}W_{s}d\langle X\rangle_{s}.$

######  Remark 2.6.

It is important to note that in the previous approximating Riemann sum, we use
the left point $t_{i}$ for $Z_{s}$ in each interval $[t_{i},t_{i+1}]$. This
special choice makes Itô’s integrals martingales.

Another important type of stochastic integration is the following.

######  Definition 2.7.

The Stratonovich integral of $Z_{t}$ against $X_{t}$ is defined to be

$\int_{0}^{t}Z_{s}\circ dX_{s}:=\int_{0}^{t}Z_{s}dX_{s}+\frac{1}{2}\langle
Z,X\rangle_{t}.$

The change of variable formula for stochastic integration is drastically
different from the classical calculus.

######  Proposition 2.8 (Itô’s formula).

Let $f\in C^{2}(\mathbb{R}^{n})$ and $\\{X_{t}\\}_{0\leq t\leq T}$ is a square
integrable martingale. Then for any $0\leq t\leq T$,

$\displaystyle f(X_{t})$
$\displaystyle=f(X_{0})+\sum_{i=1}^{n}\int_{0}^{t}\partial_{i}f(X_{s})dX^{i}_{s}+\frac{1}{2}\sum_{i,j=1}^{n}\int_{0}^{t}\partial^{2}_{i,j}f(X_{s})d\langle
X^{i},X^{j}\rangle_{s}$ (2.1)
$\displaystyle=f(X_{0})+\sum_{i=1}^{n}\int_{0}^{t}\partial_{i}f(X_{s})\circ
dX^{i}_{s}.$

We see that in (2.1), there is an extra second derivative term, which does not
appear in the classical calculus. Intuitively, this term corresponds to the
second order term in the Taylor series expansion. Since martingales have
finite quadratic variations, the sum of the second order terms in the Taylor
expansion converges to a non-zero limit, which leads to the extra term in the
Itô’s formula.

Now we can formulate the next

######  Definition 2.9.

Let $X_{t}$ be a process on $\mathbb{R}^{n}$ and $L$ a smooth second order
differential operator on $\mathbb{R}^{n}$. We say $X_{t}$ is the diffusion
generated by $L$, if for any $f\in C_{c}^{2}(\mathbb{R}^{n})$ the process

$M^{f}_{t}:=f(X_{t})-f(X_{0})+\int_{0}^{t}Lf(X_{s})ds$

is a martingale. Moreover, $L$ is called the generator of $X_{t}$.

By Itô’s formula (2.1), it is easily checked that that Brownian motion is the
diffusion generated by $\frac{1}{2}\Delta_{\mathbb{R}^{n}}$.

A canonical example of diffusion is provided by stochastic differential
equations (SDE) of the form

$X_{t}=X_{0}+\sum_{i=1}^{n}\int_{0}^{t}V_{i}(X_{s})\circ
dW^{i}_{s}+\int_{0}^{t}V_{0}(X_{s})ds,\ 0\leq t\leq T,\
X_{0}\in\mathbb{R}^{n},$ (2.2)

where $\\{V_{i}\\}_{0\leq i\leq n}$ are smooth vector fields on
$\mathbb{R}^{n}$ and $W_{t}$ is a $\mathbb{R}^{n}$-valued Brownian motion. The
following existence and uniqueness result is well-known.

######  Theorem 2.1.

Suppose $\\{V_{i}\\}_{0\leq i\leq n}$ are Lipschitz smooth vector fields on
$\mathbb{R}^{n}$. Then for any $T>0$, (2.2) admits a unique solution adapted
to the Brownian filtration $\mathcal{F}_{t}$.

In addition, the generator of $X_{t}$ defined above is given by the Hörmander
type operator $L=\frac{1}{2}\sum_{i=1}^{n}V_{i}^{2}+V_{0}$.

Diffusion are also (strong) Markov processes. More specifically, we have

$\mathbb{E}^{X_{0}}(X_{s+t}\mid\mathcal{F}_{s})=\mathbb{E}^{X_{s}}(X_{t}),\
\forall s,t\geq 0$

Here the sup-scripts are used to indicate the starting points. Intuitively, it
says that to predict the behavior of $X_{s+t}$ with the knowledge of
$\\{X_{r},\ 0\leq r\leq s\\}$ is the same as doing the prediction with
knowledge of only $X_{s}$. In other words, for $X_{t}$, given the current
position, the past and the future are independent. The Markov property of
diffusion allows one to construct semi-groups from $X_{t}$. More specifically,
for any bounded function $f\in C_{c}(\mathbb{R}^{n})$

$P_{t}f(x)=\mathbb{E}^{x}(f(X_{t}))$

is a strong continuous regular semi-group, whose generator coincides with the
generator of $X_{t}$.

###  2.2. Coupling of diffusion in Euclidean space

In this section, we review some basic facts of the coupling method of
diffusion in Euclidean space. Let $Z_{t}$ be a diffusion generated by a given
elliptic operator $L$. We have the following

######  Definition 2.10.

A couple diffusion
$(X_{t},Y_{t})\in\mathbb{R}^{n}\times\mathbb{R}^{n}\cong\mathbb{R}^{2n}$ is
said to be a coupling of $Z_{t}$, if both $X_{t}$ and $Y_{t}$ are diffusion
generated by $L$.

######  Remark 2.11.

This definition might look weird at first since one would have thought that we
should have $(X_{t},Y_{t})=(Z_{t},Z_{t})$. However, the uniqueness we claimed
for $Z_{t}$ is in the sense of probability distribution. In other words, if
$X_{t},Y_{t},Z_{t}$ have the same starting point, then for any Borel set
$B\in\mathbb{R}^{n}$, we always have

$\mathbb{P}(X_{t}\in B)=\mathbb{P}(Y_{t}\in B)=\mathbb{P}(Z_{t}\in B),\
\forall t>0$

But this does not imply $X_{t}=Y_{t}=Z_{t}$. We can give a simple example. Let
$W_{t}$ be a standard Brownian motion on $\mathbb{R}^{1}$, then by symmetry
$(W_{t},-W_{t})$ is a coupling of real Brownian motion. But clearly, $W_{t}$
and $-W_{t}$ are not identical.

For our purpose, we will consider

$\displaystyle X_{t}$
$\displaystyle=X_{0}+\sum_{i=1}^{n}\int_{0}^{t}V_{i}(X_{s})\circ
dW^{i}_{s}+\int_{0}^{t}V_{0}(X_{s})ds$ $\displaystyle Y_{t}$
$\displaystyle=Y_{0}+\sum_{i=1}^{n}\int_{0}^{t}V_{i}(Y_{s})\circ
dB^{i}_{s}+\int_{0}^{t}V_{0}(Y_{s})ds,$

where $(W_{t},B_{t})$ is a coupling of $\mathbb{R}^{n}$ Brownian motions.
Obviously, $(X_{t},Y_{t})$ is a coupling of diffusion generated by
$L=\frac{1}{2}\sum_{i=1}^{n}V_{i}^{2}+V_{0}$. So, each coupling
$(W_{t},B_{t})$ of $\mathbb{R}^{n}$ Brownian motions gives us a coupling
$(X_{t},Y_{t})$. A special type of coupling that has attracted a lot of
attentions is the so called Kendall-Cranston coupling coupling, or reflection
coupling, which was formally introduced in [5] and we now introduce.

We consider the orthogonal matrix

$m^{\prime}(x,y)=I_{n}-2\frac{(x-y)(x-y)^{*}}{\left|x-y\right|^{2}},\
x,y\in\mathbb{R}^{n},x\neq y.$

Multiplying $m^{\prime}(x,y)$ with an vector $v$ gives the reflection of $v$
across the hyperplane perpendicular to the line segment that connects $x$ and
$y$.

Let $W_{t}$ be a standard $\mathbb{R}^{n}$ Brownian motion, and define

$\displaystyle X_{t}$
$\displaystyle=X_{0}+\sum_{i=1}^{n}\int_{0}^{t}V_{i}(X_{s})\circ
dW^{i}_{s}+\int_{0}^{t}V_{0}(X_{s})ds$ (2.3) $\displaystyle Y_{t}$
$\displaystyle=Y_{0}+\sum_{i=1}^{n}\int_{0}^{t}V_{i}(Y_{s})\circ
dB^{i}_{s}+\int_{0}^{t}V_{0}(Y_{s})ds$ (2.4) $\displaystyle B_{t}$
$\displaystyle=\int_{0}^{t}m^{\prime}(X_{t},Y_{t})dW_{t}.$ (2.5)

Since $m^{\prime}(X_{s},Y_{s})$ is orthogonal, by Levy’s characterization,
$B_{t}$ is another Brownian motion adapted to $\sigma(W_{t})$. Hence, the
above-defined $(X_{t},Y_{t})$ is a coupling. We will refer to
$\tau:=\inf\\{t\geq 0:\ X_{t}=Y_{t}\\}$ as the coupling time. Now, the
complete reflection coupling is given by (2.3) - (2.5) and
$X_{t}=Y_{t},\forall t\geq\tau.$ Notice that we set $X_{t}=Y_{t}$ after the
coupling time. By the strong Markov property, this does not change the
generator of $X_{t}$ and $Y_{t}$. The fact that these two diffusion stick
together when $t$ is sufficiently large is crucial to our analysis later.

###  2.3. Diffusion on Riemannian manifolds

Let $M^{n}$ be a smooth compact Riemannian manifold of dimension $n>1$ without
boundary. We first make sense of SDE of the form

$X_{t}=X_{0}+\sum_{i=1}^{n}\int_{0}^{t}V_{i}(X_{s})\circ
dW^{i}_{s}+\int_{0}^{t}V_{0}(X_{s})ds,\ 0\leq t\leq T,\ X_{0}\in M^{n},$ (2.6)

where $\\{V_{i}\\}_{0\leq i\leq n}$ are smooth vector fields on $M^{n}$ and
$W_{t}$ is a $\mathbb{R}^{n}$-valued Brownian motion.

######  Definition 2.12.

An adapted process $X_{t}$ is said to be the solution to (2.6) if for any
$f\in C^{\infty}(M^{n})$, we have

$f(X_{t})=f(X_{0})+\sum_{i=1}^{n}\int_{0}^{t}V_{i}f(X_{s})\circ
dW^{i}_{s}+\int_{0}^{t}V_{0}f(X_{s})ds,\ 0\leq t\leq T.$

By Whitney’s embedding Theorem, $M^{n}$ can be smoothly embeded into
$\mathbb{R}^{2n}$. We can also extend $\\{V_{i}\\}_{0\leq i\leq n}$ to smooth
vector fields $\\{\hat{V}_{i}\\}_{0\leq i\leq n}$ on $\mathbb{R}^{2n}$. The
existence and uniqueness of solution to (2.6) is guaranteed by Theorem 2.1 and
the observation that $X_{t}$ as a process in $\mathbb{R}^{2n}$ stays in
$M^{n}$.

Diffusion on Riemannian manifolds is defined as a natural extension of the
Definition 2.9.

######  Definition 2.13.

Let $X_{t}$ be a process on $M^{n}$ and $L$ a smooth second order differential
operator on $M^{n}$. We say $X_{t}$ is the diffusion generated by $L$, if for
any $f\in C^{2}(M^{n})$ the process

$M^{f}_{t}:=f(X_{t})-f(X_{0})+\int_{0}^{t}Lf(X_{s})ds$

is a martingale.

We are ready for the next

######  Definition 2.14.

A process $X_{t}$ on $M^{n}$ is said to be a Brownian motion if the generator
of $X_{t}$ is $\frac{1}{2}\Delta$, where $\Delta$ is the Laplace-Beltrami
operator on $M^{n}$.

In order to take full advantage of stochastic calculus, we would like to write
a Brownian motion on $M^{n}$ as the solution to a SDE on $M^{n}$ (note that
this task is trivial on $\mathbb{R}^{n}$). Thus, it is desirable to write the
Laplace-Beltrami operator as a Hörmander type operator
$L=\frac{1}{2}\sum_{i=1}^{n}V_{i}^{2}+V_{0}$. However, there is no intrinsic
way of achieving this on a general Riemannian manifold. We will employ the
Eells-Elworthy-Malliavin approach (see chapter 3 of [10]), which we now
describe.

Let $O(M^{n})$ be the orthonormal frame bundle of $M^{n}$ and
$\pi:O(M^{n})\rightarrow M^{n}$ the canonical projection. For any $u\in
O(M^{n})$, we denote $H_{e_{i}}(u)=H_{i}(u)$ the horizontal vector field such
that $\pi_{*}H_{i}(u)=u\cdot e_{i}$, where $\\{e_{i}\\}_{1\leq i\leq n}$ is a
basis of $\mathbb{R}^{n}$. Bochner’s horizontal Laplacian on $O(M^{n})$ is
defined as $\Delta_{O(M^{n})}=\sum_{i=1}^{n}H^{2}_{i}$. In particular, we have

######  Theorem 2.2.

Let $f\in C^{\infty}(M^{n})$ and $\tilde{f}=f\circ\pi$ its lift to $O(M^{n})$,
then for any $u\in O(M^{n})$

$\Delta_{O(M^{n})}\tilde{f}(u)=\Delta f(\pi(u)).$

With Bochner’s horizontal Laplacian, a horizontal Brownian motion on
$O(M^{n})$ can be written as the solution to

$\alpha_{t}=U_{0}+\sum_{i=1}^{n}\int_{0}^{t}H_{i}(\alpha_{s})\circ
dB^{i}_{s},\ \alpha_{0}\in O(M^{n}).$

Its projection $X_{t}=\pi(\alpha_{t})$ is a Brownian motion on $M^{n}$ starts
at $\pi(\alpha_{0})$.

###  2.4. Coupling of Brownian motions on Riemannian manifolds

The definition of coupling on Riemannian manifold is exactly the same as on
$\mathbb{R}^{n}$ as we introduced before. In order to construct a reflection
coupling of Brownian motions on $M^{n}$, we only need to replace the
orthogonal matrix $m^{\prime}(x,y)$ by the so called mirror map
$m(x,y):T_{x}M^{n}\rightarrow T_{y}M^{n}$. For each $w\in T_{x}M^{n}$,
$m(x,y)(w)$ is obtained by first parallel transport $w$ along the unique
minimizing geodesic from $x$ to $y$, then reflect the resulting vector with
respect to the hyperplane perpendicular to the geodesic at $y$. Clearly,
$m(x,y)$ is an isometry and generalization of $m^{\prime}(X,Y)$ that we used
in $\mathbb{R}^{n}$.

The reflection coupling of Brownian motions on $M^{n}$ is defined as

$\displaystyle dU_{t}$ $\displaystyle=\sum_{i=1}^{n}H_{i}(U_{s})\circ
dB^{i}_{s},\ \ \ \ \ \ \ \ \ \ \ dV_{t}=\sum_{i=1}^{n}H_{i}(V_{s})\circ
dW^{i}_{s},$ $\displaystyle X_{t}$ $\displaystyle=\pi(U_{t}),\ \ \ \ \ \
Y_{t}=\pi(V_{t}),\ \ \ \ \ dW_{t}=V_{t}m(X_{t},Y_{t})U_{t}^{-1}dB_{t}.$

###  2.5. Boundary control of solutions of heat equation

We recall from [18, Lemma 3.4] a general near boundary control of positive
functions on a convex domain that vanish on the boundary .

######  Proposition 2.15.

Let $\Omega$ be a uniformly convex bounded domain in a Riemannian manifold
$M^{n}$, and $u:\overline{\Omega}\times\mathbb{R}_{+}\to\mathbb{R}$ a $C^{2}$
function such that $u$ is positive on $\Omega$, $u(\cdot,t)=0$ and $\nabla
u\neq 0$ on $\partial\Omega$. Given $T<\infty$, there exists $r_{1}>0$ such
that $\operatorname{Hess}\log u|_{(x,t)}<0$ whenever
$d(x,\partial\Omega)<r_{1}$ and $t\in[0,T]$, and $C\in\mathbb{R}$ such that
$\operatorname{Hess}\,\log u|_{(x,t)}(v,v)\leq C\|v\|^{2}$ for all
$x\in\Omega$ and $t\in[0,T]$.

We will also need the next

######  Proposition 2.16.

Let $\Omega$ be a bounded strictly convex domain in a Riemannian manifold with
smooth boundary $\partial\Omega$. Denote by
$\rho_{\partial\Omega}:\bar{\Omega}\rightarrow\mathbb{R}_{+}$the distance
function to the boundary $\partial\Omega$, and $N$ the unit inward normal
vector field on $\partial\Omega$. Let $u_{0}\in C^{\infty}(\bar{\Omega})$ be
positive in $\Omega$ such that $u_{0}=0$ and $\nabla u_{0}\neq 0$ on
$\partial\Omega$. Let
$u:\mathbb{R}_{+}\times\bar{\Omega}\rightarrow\mathbb{R}_{+}$be a smooth
solution to

$\displaystyle\frac{\partial u}{\partial t}$
$\displaystyle=(\Delta-V)u\quad\text{ in }\mathbb{R}_{+}\times\Omega;$
$\displaystyle u$ $\displaystyle=0\quad\text{ on
}\mathbb{R}_{+}\times\partial\Omega\text{ and }u(0,\cdot)=u_{0}\text{ in
}\bar{\Omega}.$

Then the solution $u$ verifies that

(1) for any $T>0,\theta_{T}:=\inf_{[0,T]\times\partial\Omega}|\nabla u|>0$;

(2) for every $x\in\partial\Omega,\nabla u(t,x)=|\nabla u(t,x)|N(x)$;

(3) $\lim_{\rho_{\partial\Omega}(x)\rightarrow 0}\frac{u(t,x)}{|\nabla
u(t,x)|\rho_{\partial\Omega}(x)}=1$ uniformly in $t\in[0,T]$;

(4) for any $T>0$, there exists $C_{T}\geq 0$ such that
$\left.\operatorname{Hess}\log u\right|_{(t,x)}(y,y)\leq C_{T}|y|^{2}$ for all
$t\in[0,T],x\in\Omega$ and $y\in T{\Omega}$ tangent vector at $x$.

###### Proof.

The first statement is a consequence of maximum principle for an elliptic
operator and non-constancy of $u_{0}$. Since $u$ behaves as a level set on
$\mathbb{R}_{+}\times\partial\Omega$, the second statement follows. The third
statement follows from the following identification: for $r>0$, we write
$\partial_{r}\Omega=\left\\{x\in\Omega:\rho_{\partial\Omega}(x)\leq
r\right\\}$ for the $r$-neighborhood of $\partial\Omega$. As [19], there
exists $r_{0}\in(0,D/2)$ such that $\rho_{\partial\Omega}$ is smooth on
$\partial_{r_{0}}\Omega$. Then for any $x\in\partial_{r_{0}}\Omega$, there
exists a unique $x^{\prime}\in\partial\Omega$ such that
$\rho_{\partial\Omega}(x)=\left|x-x^{\prime}\right|$ (meaning that the minimal
geodesic distance between $x$ and $x^{\prime}$) and
$\nabla\rho_{\partial\Omega}(x)=N\left(x^{\prime}\right)$. In particular,
$\nabla\rho_{\partial\Omega}=N$ on the boundary $\partial\Omega$. The last
statement follows from Proposition 2.15. ∎

###  2.6. Jacobi fields and Variations in the Model Spaces

Here we recall the notations and variations from [18] which we will use later.

Let $\mathbb{M}_{K}^{n}$ be the $n$-dimensional simply connected manifold with
constant sectional curvature $K$. Let $\operatorname{sn}_{K}(s)$ be the
solution of

$\operatorname{sn}_{K}^{\prime\prime}(s)+K\operatorname{sn}_{K}(s)=0,\quad\operatorname{sn}_{K}(0)=0,\quad\operatorname{sn}_{K}^{\prime}(0)=1,$

$\operatorname{cs}_{K}(s)=\operatorname{sn}_{K}^{\prime}(s)$, and
$\operatorname{tn}_{K}(s)=K\frac{\operatorname{sn}_{K}(s)}{\operatorname{cs}_{K}(s)}=-\frac{\operatorname{cs}_{K}^{\prime}(s)}{\operatorname{cs}_{K}(s)}$.
Explicitly we have

$\begin{gathered}\operatorname{sn}_{K}(s)=\begin{cases}\frac{1}{\sqrt{K}}\sin(\sqrt{K}s),&K>0\\\
s,&K=0\\\ \frac{1}{\sqrt{-K}}\sinh(\sqrt{-K}s)&K<0\end{cases}\\\
\operatorname{cs}_{K}(s)=\begin{cases}\cos(\sqrt{K}s),&K>0\\\ 1,&K=0\\\
\cosh(\sqrt{-K}s),&K<0\end{cases}\end{gathered}$

and

$\operatorname{tn}_{K}(s)=\begin{cases}\sqrt{K}\tan(\sqrt{K}s),&K>0\\\
0,&K=0\\\ -\sqrt{-K}\tanh(\sqrt{-K}s).&K<0\end{cases}$ (2.7)

Let $x_{0}\neq y_{0}$ in $\Omega$ with $\rho(x_{0},y_{0})=d_{0}$. Let
$\gamma(s):\left[-\tfrac{d_{0}}{2},\tfrac{d_{0}}{2}\right]\mapsto\Omega$ be
the normal minimal geodesic connecting $x_{0}$ to $y_{0}$. Associated to a
vector $v\oplus w\in T_{x_{0}}\Omega\oplus T_{y_{0}}\Omega$, we use the
variation $\eta(r,s)$ from [18, Page 362]. Let $\sigma_{1}(r)$ be the geodesic
with $\sigma_{1}(0)=x_{0},\frac{\partial}{\partial
r}\sigma_{1}(0)=v,\sigma_{2}(r)$ be the geodesic with
$\sigma_{2}(0)=y_{0},\frac{\partial}{\partial r}\sigma_{2}(0)=w$, and
$\eta(r,s),s\in\left[-\frac{d_{0}}{2},\frac{d_{0}}{2}\right]$, be the minimal
geodesic connecting $\sigma_{1}(r)$ and $\sigma_{2}(r)$, with
$\eta(0,s)=\gamma(s)$. Namely $\eta(r,s)=\exp_{\sigma_{1}(r)}sV(r)$ for some
$V(r)$. Since we are in a strictly convex domain, every two points are
connected by a unique minimal geodesic, the variation $\eta(r,s)$ is smooth.
Denote the variation field $\frac{\partial}{\partial r}\eta(r,s)$ by $J(r,s)$.
Then $J(r,s)$ is the Jacobi field along $s$ direction satisfying
$J\left(r,-\frac{d_{0}}{2}\right)=v,J\left(r,\frac{d_{0}}{2}\right)=w$. Denote
$J(s)=J(0,s)$. Note that with this parametrization, in general, for fixed
$r,\eta(r,s)$ is not unit speed when $s\neq 0$.

Denote the unit vector of $\frac{\partial}{\partial s}\eta(r,s)$ by

$T(r,s)=\frac{\eta^{\prime}}{\left\|\eta^{\prime}\right\|}.$ (2.8)

For space with constant sectional curvature $\mathbb{M}_{K}^{n}$, it is shown
in [18] that

$\displaystyle\left.\nabla_{r}T(r,s)\right|_{r=0}=J^{\prime}(s),$ (2.9)
$\displaystyle\left.\nabla_{r}\nabla_{r}T\right|_{r=0}=-\left\|J^{\prime}(s)\right\|^{2}e_{n}.$
(2.10)

Let $\\{e_{i}\\},i=1,\dots,n$ be an orthonormal basis of $T_{x_{0}}$ with
$e_{n}=\gamma^{\prime}(0)$, parallel translate $e_{i}$ along $\gamma$ to
$y_{0}$. In $\mathbb{M}^{n}_{k}$, we have explicit formulas for the Jacobi
fields with given boundary values. For each $1\leq i\leq n-1$, with boundary
values $0$ and $e_{i}$, the Jacobi fields are
$Q_{i}(s)=\frac{\operatorname{sn}_{K}\left(\frac{d_{0}}{2}+s\right)}{\operatorname{sn}_{K}\left(d_{0}\right)}e_{i}(s)$;
for boundary values $e_{i}$ and $0$,
$Q_{i}(s)=\frac{\operatorname{sn}_{K}\left(-s+\frac{d_{0}}{2}\right)}{\operatorname{sn}_{K}\left(d_{0}\right)}e_{i}(s)$;
and for $e_{i}$ and $e_{i}$,
$Q_{i}(s)=\frac{\operatorname{cs}_{K}\left(s\right)}{\operatorname{cs}_{K}\left(\frac{d_{0}}{2}\right)}e_{i}(s)$.

##  3\. Diffusion couplings and related properties

In this section, we introduce two different couple diffusion and derive some
useful properties for our use later. We first consider the following
$(X_{t},Y_{t})$ on the product manifold $M^{n}\times M^{n}$. This coupling
will be used to prove the gap comparison. This coupling was also used in the
Euclidean case in [8]. We consider

$\displaystyle dU_{t}=\sum_{i=1}^{n}\sqrt{2}H_{e_{i}}(U_{t})\circ
dB^{i}_{t}+2H_{\nabla\log\phi_{1}}(U_{t})dt,$ (3.1) $\displaystyle
dV_{t}=\sum_{i=1}^{n}\sqrt{2}H_{e_{i}}(V_{t})\circ
dW^{i}_{t}+2H_{\nabla\log\phi_{1}}(V_{t})dt,$ (3.2) $\displaystyle
dW_{t}=V_{t}m_{X_{t}Y_{t}}U_{t}^{-1}dB_{t},$ (3.3) $\displaystyle
X_{t}=\pi(U_{t}),\ \ \ \ \ \ Y_{t}=\pi(V_{t}),$ (3.4)

where $m(X_{t},Y_{t})$ is the mirror map defined in Section 2.4 and
$H_{e_{i}}$ is the horizontal vector field we defined in section 2.3. It is
easily checked that $(X_{t},Y_{t})$ is a diffusion coupling in the sense of
Kendall-Cranston [14, 11].

######  Remark 3.1.

The main motivation behind this definition is that both $X_{t},Y_{t}$ are
generated by the operator $L$ that appears in the ground state equation (5.2)
in section 5 below, which is essential in proving the gap comparison. It
should be pointed out that with the Neumann boundary condition, $\phi_{1}$ is
always constant. Therefore, $\nabla\phi_{1}=0$ and the coupling
$(X_{t},Y_{t})$ given above becomes a coupling of Brownian motions, which is
the canonical diffusion to study the first non-zero Neumann eigenvalue (see
[10] chapter 6).

Our first task is to prove that for arbitrary starting points
$X_{0},Y_{0}\in\Omega$, the processes $X_{t},Y_{t}$ will stay in $\Omega$ for
all $t\geq 0$.

######  Lemma 3.2.

Let $\Omega$ be a bounded strictly convex domain in a Riemannian manifold with
smooth boundary $\partial\Omega$. If $X_{0},Y_{0}\in\Omega$, then
$X_{t},Y_{t}\in\Omega$ for all $t\geq 0$.

###### Proof.

First we choose a smooth function $r:\bar{\Omega}\rightarrow\mathbb{R}$ such
that $r(x)>0$ for all $x\in\Omega$ and for some $d_{0}\in(0,D)$,

$r(x)=\rho_{\partial\Omega}(x),\ \forall
x\in\\{p\in\Omega\mid\rho_{\partial\Omega}(p)<d_{0}\\},$

where $\rho_{\partial\Omega}(x)$ denotes the geodesic distance between $x$ and
the boundary $\partial\Omega$. By our definition and compactness (see for
instance, Theorem 2.2 of [19]), it is always possible to find $c_{0}>0$, such
that $\Delta r(x)\geq-c_{0}$ for all $x\in\Omega$.

Since we only need to know the behavior of $X_{t}$ near the boundary, it
suffices to show that $r(X_{t})>0$ for all $t\geq 0$. By the same arguments in
[5, Theorem 3] or [12, Theorem 6.6.2], we have

$dr(X_{t})=\sqrt{2}d\beta_{t}+\left[\Delta
r(X_{t})+2\nabla\log\phi_{1}(r)(X_{t})\right]dt,$

where $\beta_{t}$ is a standard Brownian motion. Since
$e^{-\lambda_{1}t}\phi_{1}$ solves the heat equation in Proposition 2.16,
$\phi_{1}(x)$ verifies (2) (3) of Proposition 2.16. Hence, we can find
$d\in(0,d_{0})$ such that

$\Delta r(x)+2\nabla\log(\phi_{1})r(x)\geq
2\nabla\log(\phi_{1})r(x)-c_{0}\geq\frac{1}{r(x)},$

for all $x\in\Omega$ with $\rho_{\partial\Omega(x)}<d$. So we get, when
$\rho_{\partial\Omega}(X_{t})<d$,

$dr(X_{t})>\sqrt{2}d\beta_{t}+\frac{1}{r(X_{t})}dt,$

which implies $r(X_{t})>0$. Our proof is complete. ∎

Recall that $\tau$ is the coupling time of $(X_{t},Y_{t})$. Our next result
shows that $\tau$ is finite almost surely.

######  Lemma 3.3.

Let $(M^{n},g)$ be a complete Riemannian manifold with Ricci curvature lower
bound $(n-1)k,k\in\mathbb{R}$ and $\Omega\subset M^{n}$ be a convex domain
with diameter $D>0$. Then, for any starting points $X_{0},Y_{0}\in\Omega$, the
coupling $(X_{t},Y_{t})$ is successful in the sense that $\tau<\infty$ almost
surely.

###### Proof.

Let $\rho_{t}=\rho(X_{t},Y_{t})$ be the geodesic distance between
$X_{t},Y_{t}$. It suffices to prove $\rho_{t}$ hits $0$ in finite time almost
surely. Let $\\{e_{i}\\}_{1\leq i\leq n}$ be an orthonormal frame at $X_{t}$
and we parallel transport them along the unique minimizing geodesic that
connects $X_{t},Y_{t}$. We further assume that $e_{n}$ is the tangential
direction. By Itô’s formula (see section 6.6 of [11] for a similar
computation)

$d\rho_{t}=2\sqrt{2}d\beta_{t}+\left(\sum_{i=1}^{n}I(Q_{i},Q_{i})+2\langle\nabla\log\phi_{1}(Y_{t}),\gamma^{\prime}(Y_{t})\rangle-2\langle\nabla\log\phi_{1}(X_{t}),\gamma^{\prime}(X_{t})\rangle\right)dt,$

where $\beta_{t}$ is a Brownian motion independent from $B_{t}$ and $Q_{i}$ is
the Jacobi field associated with any variation whose tangent vector at
$(X_{t},Y_{t})$ is given by $e_{i}\oplus m(X_{t},Y_{t})e_{i}$. The index form
of a vector field $X$ (with not necessarily vanishing endpoints) along a
geodesic $\gamma$ is defined by

$\displaystyle
I(\gamma,X,X):=\int_{0}^{T}\left(\langle\nabla_{\gamma^{\prime}}X,\nabla_{\gamma^{\prime}}X\rangle-\langle
R(\gamma^{\prime},X)X,\gamma^{\prime}\rangle\right)dt,$

where $\nabla$ is the Levi-Civita connection and $R$ is the Riemann curvature
tensor of a given manifold. We will use Lemma 1 from [15] to verify that
$\rho_{t}$ hits $0$ in finite time. We have

$\displaystyle\theta_{t}:$
$\displaystyle=2\left(\langle\nabla\log\phi_{1}(Y_{t}),\gamma^{\prime}(Y_{t})\rangle-\langle\nabla\log\phi_{1}(X_{t}).\gamma^{\prime}(X_{t})\rangle\right)+\sum_{i=1}^{n}I(Q_{i},Q_{i})$
$\displaystyle=2\int_{0}^{\rho_{t}}\nabla^{2}\log\phi_{1}(\gamma^{\prime}(s),\gamma^{\prime}(s))ds+\sum_{i=1}^{n}I(Q_{i},Q_{i}).$

By Index lemma, we can find $C_{1}(k)>0$ such that

$\frac{1}{2}\sum_{i=1}^{n}I(Q_{i},Q_{i})\leq-(n-1)tn_{k}(\rho_{t}/2)\leq
C_{1}.$

Consequently

$\theta_{t}\leq
2\int_{0}^{\rho_{t}}\nabla^{2}\log\phi_{1}(\gamma^{\prime}(s),\gamma^{\prime}(s))ds+C_{1}.$

Let

$C_{2}=\sup\\{\operatorname{Hess}\phi_{1}(p)(v,v):p\in\bar{\Omega},\ v\in
T_{p}M^{n},\ |v|=1\\}.$

By (4) of Proposition 2.16, $C_{2}$ is finite. We introduce

$\eta(r)=\begin{cases}\frac{1}{4}C_{2}\cdot r+C_{1}&r\leq D\\\
-1&r>D.\end{cases}$

Since $X_{t},Y_{t}$ never leave $\Omega$, $\rho_{t}\leq D$ for all $t\geq 0$
and we have $\theta_{t}\leq 8\eta(\rho_{t})$ for all $t\geq 0$. It follows
that the assumptions of Lemma 1 of [15] are verified. We conclude that
$\rho_{t}$ hits $0$ in finite time almost surely. The proof is complete. ∎

Next we introduce another important pair of diffusion
$(X^{\prime}_{t},Y^{\prime}_{t})$, which will be used in establishing the log-
concavity estimate for the first eigenfunction of $-\Delta+V$. Let us consider

$\displaystyle
dU^{\prime}_{t}=\sum_{i=1}^{n}\sqrt{2}H_{e_{i}}(U^{\prime}_{s})\circ
dB^{i}_{s}+2H_{\nabla\log\phi_{1}}(U^{\prime}_{t})+2tn_{k}\left(\frac{\tilde{\rho}(U^{\prime}_{t},V^{\prime}_{t})}{2}\right)\tilde{\gamma}^{\prime}(U^{\prime}_{s})ds,$
(3.5) $\displaystyle
dV^{\prime}_{t}=\sum_{i=1}^{n}\sqrt{2}H_{e_{i}}(V^{\prime}_{s})\circ
dW^{i}_{s}+2H_{\nabla\log\phi_{1}}(V^{\prime}_{t})-2tn_{k}\left(\frac{\tilde{\rho}(U^{\prime}_{t},V^{\prime}_{t})}{2}\right)\tilde{\gamma}^{\prime}(V^{\prime}_{s})ds,$
$\displaystyle
dW_{t}=V^{\prime}_{t}m_{X_{t}Y_{t}}(U^{\prime}_{t})^{-1}dB_{t},$
$\displaystyle X^{\prime}_{t}=\pi(U^{\prime}_{t}),\ \ \ \ \ \
Y^{\prime}_{t}=\pi(V^{\prime}_{t}).$

Here $\tilde{\rho}$ is the lift of $\rho$ and $\tilde{\gamma}$ is the
horizontal lift of the minimizing geodesic that goes from $X^{\prime}_{t}$ to
$Y^{\prime}_{t}$. We note that $X^{\prime}_{t},Y^{\prime}_{t}$ can not be
defined separately since the drift term depends on both $X^{\prime}_{t}$ and
$Y^{\prime}_{t}$. For this reason, $(X^{\prime}_{t},Y^{\prime}_{t})$ is not a
coupling of diffusion. We define $\tau^{\prime}=\inf_{t\geq
0}\\{\rho(X^{\prime}_{t},Y^{\prime}_{t})=0$}. With a slight abuse of
terminology, we will still call $\tau^{\prime}$ the coupling time. As before,
we define $X^{\prime}_{t}=Y^{\prime}_{t}$ when $t\geq\tau^{\prime}$.

######  Remark 3.4.

There is no intuitive explanation of the extra term, compared to
$(X_{t},Y_{t})$, added in the definition of $(X^{\prime}_{t},Y^{\prime}_{t})$;
it was what came out of our computations. However, it is interesting to note
that when $k=0$, the couple diffusion $(X^{\prime}_{t},Y^{\prime}_{t})$
coincides with the coupling $(X_{t},Y_{t})$. We can thus regard
$(X^{\prime}_{t},Y^{\prime}_{t})$ as a non-trivial generalization of
$(X_{t},Y_{t})$ that has incorporated more information about the curvature of
the underlying manifold.

One can argue in the same way as we did in Lemma 3.2 to justify the next

######  Lemma 3.5.

Under the same assumptions of Lemma 3.2, if
$X^{\prime}_{0},Y^{\prime}_{0}\in\Omega$, then
$X^{\prime}_{t},Y^{\prime}_{t}\in\Omega$ for all $t\geq 0$.

Let us observe that, $(X_{t}^{\prime},Y_{t}^{\prime})$, although do not form a
usual coupling, will also meet in finite time. In particular, we can give a
comparison between $\tau$ and $\tau^{\prime}$ based on $k$.

######  Lemma 3.6.

As in the setting of Lemma 3.3, we have $\tau^{\prime}<\infty$ almost surely.
Moreover, we have $\tau^{\prime}\leq\tau$ when $k\geq 0$ and
$\tau>\tau^{\prime}$ when $k<0$ almost surely.

###### Proof.

Since
$\nabla_{\gamma^{\prime}(X^{\prime}_{t})}\rho(X^{\prime}_{t},Y^{\prime}_{t})=-\nabla_{\gamma^{\prime}(Y^{\prime}_{t})}\rho(X^{\prime}_{t},Y^{\prime}_{t})=-1$
, we have by Itô’s formula

$\displaystyle
d\rho(X^{\prime}_{t},Y^{\prime}_{t})=2\sqrt{2}d\beta_{t}+\left(\sum_{i=1}^{n}I(Q_{i},Q_{i})+F(X^{\prime}_{t},Y^{\prime}_{t})-4tn_{k}\left(\frac{\rho(X^{\prime}_{t},Y^{\prime}_{t})}{2}\right)\right)dt,$

where

$F(X_{t},Y_{t})=\langle\nabla\log\phi_{1}(Y_{t}),\gamma^{\prime}(Y_{t})\rangle-\langle\nabla\log\phi_{1}(X_{t}),\gamma^{\prime}(X_{t})\rangle,$

and $\beta_{t}$ is a standard Brownian motion. Since
$tn_{k}\left(\rho(X^{\prime}_{t},Y^{\prime}_{t})/2\right)$ is uniformly
bounded in $\Omega$, the exact same argument of Lemma 3.3 applies to
$\rho(X^{\prime}_{t},Y^{\prime}_{t})$ and $\tau^{\prime}<+\infty$ follows.

For our second claim, we observe that
$tn_{k}\left(\rho(X^{\prime}_{t},Y^{\prime}_{t})/2\right)\geq 0$ when $k\geq
0$ and $tn_{k}\left(\rho(X^{\prime}_{t},Y^{\prime}_{t})/2\right)<0$ when
$k<0$; from which and the diffusion comparison theorem we deduce that
$\rho(X_{t},Y_{t})\geq\rho(X^{\prime}_{t},Y^{\prime}_{t})$ when $k\geq 0$ and
$\rho(X_{t},Y_{t})<\rho(X^{\prime}_{t},Y^{\prime}_{t})$ when $k<0$. The proof
is complete. ∎

##  4\. Log-concavity of the first eigenfunction

In this section we establish a log-concavity estimate of the first
eigenfunction of the Schrödinger operator $-\Delta+V$ on $\Omega$, Theorem
1.1. Our analysis in this section rely on the diffusion
$(X^{\prime}_{t},Y^{\prime}_{t})$ that we introduced in (3.5).

We pick two distinct points $x,y\in\Omega$ and let them be the starting points
of $X^{\prime}_{t},Y^{\prime}_{t}$ respectively. Denote
$\xi_{t}=\rho(X^{\prime}_{t},Y^{\prime}_{t})/2$. Let $\gamma$ be the normal
minimal geodesic that goes from $X^{\prime}_{t}$ to $Y^{\prime}_{t}$ with
$\gamma(-\xi_{t})=X_{t}^{\prime}$ and $\gamma(\xi_{t})=Y_{t}^{\prime}$. We
choose $\\{e_{i}\\}_{1\leq i\leq n}$ an orthonormal basis at $X_{t}$ with
$e_{n}=\gamma^{\prime}(X^{\prime}_{t})$ and parallel transport
$\\{e_{i}\\}_{1\leq i\leq n}$ along $\gamma$. Let

$E_{i}=e_{i}\oplus e_{i},\ 1\leq i\leq n-1,\ E_{n}=e_{n}\oplus-e_{n}.$

We define

$F_{t}=\langle\nabla\log\phi_{1}(Y^{\prime}_{t}),\gamma^{\prime}(Y^{\prime}_{t})\rangle-\langle\nabla\log\phi_{1}(X^{\prime}_{t}),\gamma^{\prime}(X^{\prime}_{t})\rangle.$
(4.1)

One crucial step of our method is to find the SDE of $F_{t}$. The essence is
to use Itô’s formula and the geodesic variations as in [18] (see Section 2.6)
to compute the first two orders of covariant derivatives.

The following is our key formula.

######  Proposition 4.1.

Let $\lambda_{1}$ be the first eigenvalue of $-\Delta+V$ on a convex domain
$\Omega\subset\mathbb{M}^{n}_{k}$ and $\omega=\log\phi_{1}$. Then the SDE of
$F_{t}$ is given by

$\displaystyle dF_{t}$ $\displaystyle=d\\{\text{martingale}\\}$
$\displaystyle+\left\\{\langle\nabla
V(Y^{\prime}_{t}),e_{n}\rangle-\langle\nabla
V(X^{\prime}_{t}),e_{n}\rangle\right\\}dt$
$\displaystyle+(n-1)(K-tn^{2}_{k}(\xi_{t}))\left\\{\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle\right\\}dt$
$\displaystyle+2tn_{k}(\xi_{t})\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle^{2}+\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle^{2}+2\lambda_{1}-V(X_{t}^{\prime})-V(Y_{t}^{\prime})\right)dt$
$\displaystyle+\frac{2}{sn_{k}(2\xi_{t})}\sum_{i=1}^{n-1}\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle-\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\right)^{2}dt.$
(4.2)

######  Remark 4.2.

It is possible to give the precise expression for the martingale part of
$dF_{t}$, which, however, is irrelevant to our purpose. We write it as it is
now to save space.

###### Proof.

Recall we use the notation $\xi_{t}=\rho(X^{\prime}_{t},Y^{\prime}_{t})/2$. By
Itô’s formula (2.1), we have

$\displaystyle dF_{t}$ $\displaystyle=\sum_{i=1}^{n}\nabla_{e_{i}\oplus
0}F_{t}(X^{\prime}_{t},Y^{\prime}_{t})\circ
d(X^{\prime}_{t})^{i}+\nabla_{0\oplus
e_{i}}F_{t}(X^{\prime}_{t},Y^{\prime}_{t})\circ d(Y^{\prime}_{t})^{i}$
$\displaystyle=d\\{\text{martingale}\\}+\\{\sum_{i=1}^{n}\nabla_{E_{i}}\nabla_{E_{i}}F_{t}(X^{\prime}_{t},Y^{\prime}_{t})\\}dt$
$\displaystyle+\left\\{2\nabla\log\phi_{1}(X^{\prime}_{t})(F_{t})+2\nabla\log\phi_{1}(Y^{\prime}_{t})(F_{t})+2tn_{k}\left(\xi_{t}\right)\nabla_{e_{n}\oplus
0}F_{t}-2tn_{k}\left(\xi_{t}\right)\nabla_{0\oplus e_{n}}F_{t}\right\\}dt.$

We begin with first derivative terms. Use the variation from Section 2.6 for
the vector $\nabla\omega(X_{t}^{\prime})\oplus 0$, denote $\eta(r,s)$ the
variation, and $T$ the unit variation field, see (2.8).

$\displaystyle\nabla\log\phi_{1}(X^{\prime}_{t})(F_{t})$ $\displaystyle=$
$\displaystyle\langle\nabla\omega(Y^{\prime}_{t}),\nabla_{\tfrac{\partial}{\partial
r}}T(r,s)\rangle|_{r=0,s=\xi_{t}}-\left(\langle\nabla_{\nabla\omega}\nabla\omega(r,s),T(r,s)\rangle+\langle\nabla\omega(X^{\prime}_{t}),\nabla_{\tfrac{\partial}{\partial
r}}T(r,s)\rangle\right)|_{r=0,s=-\xi_{t}}$

By [18, Page 363]

$\nabla_{\tfrac{\partial}{\partial
r}}T(r,s)|_{r=0}=-\langle\gamma^{\prime}(s),J^{\prime}(s)\rangle
e_{n}+J^{\prime}(s),$

where $J(s)$ is the Jacobi field along $\gamma$ with
$J(-\xi_{t})=\nabla\omega(X_{t}^{\prime}),\ J(\xi_{t})=0$. For
$\mathbb{M}_{k}^{n}$,

$J(s)=\left(-\frac{\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle}{2\xi_{t}}s+\frac{\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle}{2}\right)e_{n}+\sum_{i=1}^{n-1}\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\frac{sn_{k}(\xi_{t}-s)}{sn_{k}(2\xi_{t})}e_{i}.$

Hence

$J^{\prime}(s)=-\frac{\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle}{2\xi_{t}}e_{n}-\sum_{i=1}^{n-1}\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\frac{cn_{k}(\xi_{t}-s)}{sn_{k}(2\xi_{t})}e_{i},$
$\nabla_{\tfrac{\partial}{\partial
r}}T(r,s)|_{r=0}=-\sum_{i=1}^{n-1}\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\frac{cn_{k}(\xi_{t}-s)}{sn_{k}(2\xi_{t})}e_{i}.$

Plug these in, we have

$\displaystyle\nabla\log\phi_{1}(X^{\prime}_{t})(F_{t})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle\langle\nabla\omega(X_{t}^{\prime}),e_{i}\rangle\frac{-1}{sn_{k}(2\xi_{t})}+\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle^{2}\frac{cn_{k}(2\xi_{t})}{sn_{k}(2\xi_{t})}\right)$
$\displaystyle-\langle\nabla_{\nabla\omega}\nabla\omega(r,s),\gamma^{\prime}(X_{t}^{\prime})\rangle|_{r=0,s=-\xi_{t}}.$

Similarly

$\displaystyle\nabla\log\phi_{1}(Y^{\prime}_{t})(F_{t})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle\langle\nabla\omega(X_{t}^{\prime}),e_{i}\rangle\frac{-1}{sn_{k}(2\xi_{t})}+\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle^{2}\frac{cn_{k}(2\xi_{t})}{sn_{k}(2\xi_{t})}\right)$
$\displaystyle+\langle\nabla_{\nabla\omega}\nabla\omega(r,s),\gamma^{\prime}(Y_{t}^{\prime})\rangle|_{r=0,s=\xi_{t}}.$

Together with the fact that
$\nabla_{\nabla\omega}\nabla\omega=\frac{1}{2}\nabla\|\nabla\omega\|^{2}$, we
get

$\displaystyle
2\nabla\log\phi_{1}(X^{\prime}_{t})(F_{t})+2\nabla\log\phi_{1}(Y^{\prime}_{t})(F_{t})$
$\displaystyle=\langle\nabla\|\nabla\omega(Y^{\prime}_{t})\|^{2},\gamma^{\prime}(Y^{\prime}_{t})\rangle-\langle\nabla\|\nabla\omega(X^{\prime}_{t})\|^{2},\gamma^{\prime}(X^{\prime}_{t})\rangle$
$\displaystyle+2\sum_{i=1}^{n-1}\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle\left(\frac{cs_{k}(2\xi_{t})}{sn_{k}(2\xi_{t})}\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle-\frac{1}{sn_{k}(2\xi_{t})}\langle\nabla\omega(X^{\prime}_{s}),e_{i}\rangle\right)$
$\displaystyle-2\sum_{i=1}^{n-1}\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\left(\frac{1}{sn_{k}(2\xi_{t})}\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle-\frac{cs_{k}(2\xi_{t})}{sn_{k}(2\xi_{t})}\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\right).$
(4.3)

For the other first order term, we have

$\displaystyle 2tn_{k}\left(\xi_{t}\right)\nabla_{e_{n}\oplus
0}F_{t}-2tn_{k}\left(\xi_{t}\right)\nabla_{0\oplus e_{n}}F_{t}$
$\displaystyle=-2tn_{k}\left(\xi_{t}\right)\langle\nabla_{e_{n}}\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-2tn_{k}\left(\xi_{t}\right)\langle\nabla_{e_{n}}\nabla\omega(X^{\prime}_{t}),e_{n}\rangle.$
(4.4)

For the second order derivative, we argue in the exact same way as equation
(3.18) of [18] and get

$\displaystyle\sum_{i=1}^{n}\nabla_{E_{i}}\nabla_{E_{i}}F_{t}(X^{\prime}_{t},Y^{\prime}_{t})=\langle\Delta\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\Delta\nabla\omega(X^{\prime}_{t}),e_{n}\rangle$
$\displaystyle-$ $\displaystyle
2tn_{k}(\xi_{t})\sum_{i=1}^{n-1}\left(\langle\nabla_{e_{i}}\nabla\omega(Y^{\prime}_{s}),e_{i}\rangle+\langle\nabla_{e_{i}}\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\right)$
$\displaystyle-$
$\displaystyle(n-1)tn^{2}_{k}(\xi_{t})\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle\right)$
$\displaystyle=$
$\displaystyle\langle\Delta\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\Delta\nabla\omega(X^{\prime}_{t}),e_{n}\rangle$
$\displaystyle-$ $\displaystyle
2tn_{k}(\xi_{t})\left(\Delta\omega(Y^{\prime}_{t})+\Delta\omega(X^{\prime}_{t})\right)+2tn_{k}\left(\xi_{t}\right)\langle\nabla_{e_{n}}\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle+2tn_{k}\left(\xi_{t}\right)\langle\nabla_{e_{n}}\nabla\omega(X^{\prime}_{t}),e_{n}\rangle$
$\displaystyle-$
$\displaystyle(n-1)tn^{2}_{k}(\xi_{t})\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle\right).$
$\displaystyle=$
$\displaystyle\langle\Delta\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\Delta\nabla\omega(X^{\prime}_{t}),e_{n}\rangle$
$\displaystyle+$ $\displaystyle
2tn_{k}(\xi_{t})\left(2\lambda_{1}+\|\omega(Y^{\prime}_{t})\|^{2}+\|\omega(X^{\prime}_{t})\|^{2}-V(X_{t}^{\prime})-V(Y_{t}^{\prime})\right)$
$\displaystyle+$ $\displaystyle
2tn_{k}\left(\xi_{t}\right)\langle\nabla_{e_{n}}\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle+2tn_{k}\left(\xi_{t}\right)\langle\nabla_{e_{n}}\nabla\omega(X^{\prime}_{t}),e_{n}\rangle$
$\displaystyle-$
$\displaystyle(n-1)tn^{2}_{k}(\xi_{t})\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle\right).$
(4.5)

Here, we used $\Delta\omega=-\lambda_{1}-\|\nabla\omega\|^{2}+V$ in the last
equation. Finally, note that
$\nabla\|\nabla\omega\|^{2}=-\nabla\Delta\omega+\nabla V$. Then, equation
(4.2) follows from adding up (4.3) (4.4) (4.5), the Bochner–Weitzenböck
formula

$\Delta\nabla\omega-\nabla\Delta\omega=Ric(\nabla\omega,\cdot)=(n-1)k\langle\nabla\omega,\cdot\rangle,$

and the identity
$tn_{k}(\xi_{t})=\frac{1-cs_{k}(2\xi_{t})}{sn_{k}(2\xi_{t})}$. ∎

Now we are ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Since adding a constant to the linear operator $-\Delta+V$ does not change the
fundamental gap, we can assume without loss of generality that
$V\geq\sup_{[0,\tfrac{D}{2}]}\tilde{V}$.

First, we pick $0<D<D^{\prime}<\pi/\sqrt{k}$ and continuously extend
$\tilde{V}$ to an even function on $[-D^{\prime}/2,D^{\prime}/2]$. Let
$\bar{\phi}_{1}$ be the first eigenfunction of the 1-dimensional model on
$[-D^{\prime}/2,D^{\prime}/2]$. Moreover, let $\\{Q_{i}\\}_{1\leq i\leq n}$ be
the Jacobi fields of the geodesic variations $\eta(r,s)$ associated with the
vector fields $\\{E_{i}\\}_{1\leq i\leq n}$, which we introduced at the
beginning of this section. For simplicity, we denote
$\Psi=(\log\bar{\phi}_{1})^{\prime}$.

By Itô’s formula (2.1) and the second variation formula,

$\displaystyle d\Psi(\xi_{t})$ $\displaystyle=d\\{\text{martingale}\\}$
$\displaystyle+\\{\frac{1}{2}\sum_{i=1}^{n}I(Q_{i},Q_{i})\Psi^{\prime}+F_{t}\Psi^{\prime}+\Psi^{\prime\prime}-2tn_{k}\left(\xi_{t}\right)\Psi^{\prime}\\}dt.$
(4.6)

For $\mathbb{M}^{n}_{k}$, we have

$\frac{1}{2}\sum_{i=1}^{n}I(Q_{i},Q_{i})=-(n-1)tn_{k}(\xi_{t}).$ (4.7)

On the other hand, a direct computation (similar to Lemma 2.7 of [18]) gives
that

$\Psi^{\prime\prime}+2\Psi^{\prime}\Psi-
tn_{k}\left((n+1)\Psi^{\prime}+2\bar{\lambda}_{1}+2\Psi^{2}+2\tilde{V}\right)-\tilde{V}^{\prime}-(n-1)(K-tn_{k}^{2})\Psi=0.$
(4.8)

Combining estimates (4.8)(4.7) with (4.2) (4.6) gives

$\displaystyle d\left(F_{t}-2\Psi(\xi_{t})\right)$
$\displaystyle=d\\{\text{martingale}\\}+4tn_{k}\,(\lambda_{1}-\bar{\lambda}_{1})$
$\displaystyle+\left((n-1)(K-tn_{k}^{2})-2\Psi^{\prime}\right)(F_{t}-2\Psi)-4tn_{k}(\xi_{t})\Psi^{2}dt$
$\displaystyle+\left\\{\langle\nabla
V(Y^{\prime}_{t}),e_{n}\rangle-\langle\nabla
V(X^{\prime}_{t}),e_{n}\rangle-2\tilde{V}^{\prime}(\xi_{t})\right\\}-2tn_{k}(\xi_{t})(2\tilde{V}(\xi_{t})-V(X^{\prime}_{t})-V(Y^{\prime}_{t}))dt$
$\displaystyle+2tn_{k}(\xi_{t})\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle^{2}+\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle^{2}\right)dt$
$\displaystyle+\frac{2}{sn_{k}(2\xi_{t})}\sum_{i=1}^{n-1}\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{i}\rangle-\langle\nabla\omega(X^{\prime}_{t}),e_{i}\rangle\right)^{2}dt$
$\displaystyle\geq
d\\{\text{martingale}\\}+\left((n-1)(K-tn_{k}^{2}(\xi_{t}))-2\Psi^{\prime}\right)(F_{t}-2\Psi)dt$
$\displaystyle+tn_{k}(\xi_{t})(F_{t}+2\Psi)(F_{t}-2\Psi)dt$
$\displaystyle=d\\{\text{martingale}\\}$
$\displaystyle+\left((n-1)(K-tn_{k}^{2}(\xi_{t}))-2\Psi^{\prime}+tn_{k}(\xi_{t})(F_{t}+2\Psi)\right)(F_{t}-2\Psi)dt.$
(4.9)

Here we used the fact that $\tilde{V}$ is a modulus of convexity of $V$,
$V\geq\tilde{V}$, $k\geq 0$, $\lambda_{1}\geq\bar{\lambda}_{1}$ (the proof of
Lemma 3.12 in [18] still holds in our case as $V\geq\sup\tilde{V}$), and

$\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle^{2}+\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle^{2}\geq\frac{\left(\langle\nabla\omega(Y^{\prime}_{t}),e_{n}\rangle-\langle\nabla\omega(X^{\prime}_{t}),e_{n}\rangle\right)^{2}}{2}=\frac{F_{t}^{2}}{2}$

in the second inequality above. Next, we observe that (4.9) is equivalent to

$\displaystyle
d\left(e^{\int_{0}^{t}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}(F_{t}-2\Psi(\xi_{t}))\right)$
$\displaystyle\geq
e^{\int_{0}^{t}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}d\\{\text{martingale}\\}.$

Let $N>0$, integrating from $0$ to $\tau^{\prime}\wedge N:=\min\\{\tau,N\\}$
gives

$\left(e^{\int_{0}^{\tau^{\prime}\wedge
N}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}(F_{\tau^{\prime}\wedge
N}-2\Psi(\xi_{\tau^{\prime}\wedge
N})\right)\geq\left(F_{0}-2\Psi(\xi_{0})\right)+\text{Martingale}_{\tau^{\prime}\wedge
N}.$

Since $\tau^{\prime}\wedge N$ is bounded, the stopped martingale
$\text{Martingale}_{\tau^{\prime}\wedge N}$ in the above inequality is another
martingale. We can thus take expectation and get

$\displaystyle F_{0}-2\Psi(\xi_{0})$
$\displaystyle\leq\mathbb{E}\left(e^{\int_{0}^{\tau^{\prime}\wedge
N}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}(F_{\tau^{\prime}\wedge
N}-2\Psi(\xi_{\tau^{\prime}\wedge N})\right)$
$\displaystyle\leq\mathbb{E}\left(e^{\int_{0}^{\tau^{\prime}\wedge
N}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}\left|F_{\tau^{\prime}\wedge
N}-2\Psi(\xi_{\tau^{\prime}\wedge N})\right|\right).$

By Fatou,

$F_{0}-2\Psi(\xi_{0})\leq\mathbb{E}\left(\liminf_{N\rightarrow\infty}e^{\int_{0}^{\tau^{\prime}\wedge
N}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}\left|F_{\tau^{\prime}\wedge
N}-2\Psi(\xi_{\tau^{\prime}\wedge N})\right|\right).$

Since $\tau^{\prime}<\infty$ almost surely by lemma 3.6, we have
$\tau^{\prime}\wedge N\rightarrow\tau^{\prime}$ as $N$ approaches infinity. On
the other hand, $F_{\tau^{\prime}}=\Psi(\xi_{\tau^{\prime}})=0$. Thus,

$\displaystyle\mathbb{E}\left(\liminf_{N\rightarrow\infty}e^{\int_{0}^{\tau^{\prime}\wedge
N}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}\left|F_{\tau^{\prime}\wedge
N}-2\Psi(\xi_{\tau^{\prime}\wedge N})\right|\right)$
$\displaystyle=\mathbb{E}\left(e^{\int_{0}^{\tau^{\prime}}2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{s}))-tn_{k}(\xi_{s})(F_{s}+2\Psi)ds}\left|F_{\tau^{\prime}}-2\Psi(\xi_{\tau^{\prime}})\right|\right)=0.$

We conclude that

$F_{0}-2\Psi(\xi_{0})=\langle\nabla\log\phi_{1}(y),\gamma^{\prime}(y)\rangle-\langle\nabla\log\phi_{1}(x),\gamma^{\prime}(x)\rangle-2\Psi(\rho(x,y)/2)\leq
0.$

Finally, we send $D^{\prime}\rightarrow D$. The proof is complete. ∎

######  Remark 4.3.

Our proof should be compared with that of Theorem 3.2 of [8]. In particular,
we achieved two important improvements besides the handling of all the
curvature terms. First, we did not assume the potential $V$ is convex. Our
proof with optional stopping technique does not require $\Psi^{\prime}$ to be
non-positive (see remark 3.3 of [8]). Second, our argument does not require
the first eigenfunction $\phi_{1}$ to be log-concave, which was crucial in the
proof in [8].

######  Remark 4.4.

It is worth mentioning that at the critical point where
$F_{t}=2\Psi(\xi_{t})$, the quantity
$2\Psi^{\prime}-(n-1)(K-tn_{k}^{2}(\xi_{t}))-tn_{k}(\xi_{t})(F_{t}+2\Psi)$,
which appears in (4.9) and later as the integrand in the exponential function
becomes $2\Psi^{\prime}-4tn_{k}(\xi_{t})\Psi-(n-1)(K-tn_{k}^{2}(\xi_{t}))$.
The latter appeared in Theorem 3.8 of [18] (see also Remark 3.9 in the same
reference) and was used in the proof of Theorem 1.6 of [18]. The diameter
restriction $D<\pi/(2\sqrt{k})$ was coming from the need to make this quantity
non-positive. In our proof above, this sign restriction, and hence the
diameter restriction, was removed thanks to the optional stopping technique.

##  5\. Fundamental gap comparison

In this section, we prove Theorem 1.2. Let $\phi_{1},\ \phi_{2}$ be the
eigenfunctions associated with the first two eigenvalues of the Schrödinger
operator $-\Delta+V$ on $\Omega$ respectively. Recall that the ground state
transform

$v_{t}=\frac{e^{\lambda_{2}t}\phi_{2}}{e^{\lambda_{1}t}\phi_{1}}=e^{-(\lambda_{2}-\lambda_{1})t}\frac{\phi_{2}}{\phi_{1}}$
(5.1)

is a smooth solution to

$\displaystyle\partial_{t}v$ $\displaystyle=Lv=\Delta
v+2\langle\nabla\log\phi_{1},\nabla v\rangle$ (5.2) $\displaystyle v(0,\cdot)$
$\displaystyle=\phi_{2}/\phi_{1}$

on $\mathbb{R}^{+}\times\bar{\Omega}$. We will use the coupling $X_{t},Y_{t},$
from (3.1)-(3.4).

###### Proof of Theorem 1.2.

Let $\rho_{t}=\rho(X_{t},Y_{t})$ and $\xi_{t}=\rho_{t}/2$. By Itô’s formula,

$d\rho_{t}=2\sqrt{2}d\beta_{t}+\left(\sum_{i=1}^{n}I(Q_{i},Q_{i})+2\langle\nabla\log\phi_{1}(Y_{t}),\gamma^{\prime}(Y_{t})\rangle-2\langle\nabla\log\phi_{1}(X_{t}),\gamma^{\prime}(X_{t})\rangle\right)dt,$
(5.3)

where $\beta_{t}$ is a Brownian motion independent from $B_{t}$. Let
$\tilde{\phi}_{1},\tilde{\phi}_{2}$ be the first two eigenfunctions of the
corresponding one-dimensional model. Define
$\Phi=\tilde{\phi}_{2}/\tilde{\phi}_{1}$ and let

$F_{t}=\langle\nabla\log\phi_{1}(Y_{t}),\gamma^{\prime}(Y_{t})\rangle-\langle\nabla\log\phi_{1}(X_{t}),\gamma^{\prime}(X_{t})\rangle.$

With (5.3) at hand, we apply Itô’s formula again and get

$\displaystyle
d\Phi(\xi_{t})=\sqrt{2}\Phi^{\prime}(\xi_{t})d\beta_{t}+\left(\frac{1}{2}\Phi^{\prime}(\xi_{t})\sum_{i=1}^{n}I(Q_{i},Q_{i})+\Phi^{\prime}(\xi_{t})F_{t}+\Phi^{\prime\prime}(\xi_{t})\right)dt.$

By our assumption, $\log\tilde{\phi}_{1}$ is a modulus of concavity of
$\log\phi_{1}$. Thus, we have $F_{t}\leq
2(\log\tilde{\phi}_{1})^{\prime}(\xi_{t})$. Moreover, by Index lemma

$\frac{1}{2}\sum_{i=1}^{n}I(Q_{i},Q_{i})\leq-(n-1)tn_{k}(\xi_{t}).$

As a result,

$\displaystyle
d\Phi(\xi_{t})\leq\sqrt{2}\Phi^{\prime}(\xi_{t})d\beta_{t}+\left(-\Phi^{\prime}(\xi_{t})(n-1)tn_{k}(\xi_{t})+\Phi^{\prime}(\xi_{t})2(\log\tilde{\phi}_{1})^{\prime}(\xi_{t})+\Phi^{\prime\prime}(\xi_{t})\right)dt.$
(5.4)

It is easily checked that equation (2.7) of [18] still holds for the
Schrödinger operator $-\Delta+V$. Hence, we have

$\Phi^{\prime\prime}(s)-\Phi^{\prime}(s)(n-1)tn_{k}(s)+\Phi^{\prime}(s)2(\log\tilde{\phi}_{1})^{\prime}(s)=-(\bar{\lambda}_{2}-\bar{\lambda}_{1})\Phi(s).$
(5.5)

Combining (5.4) and (5.5) gives

$d\Phi(\xi_{t})\leq\sqrt{2}\Phi^{\prime}(\xi_{t})d\beta_{t}-(\bar{\lambda}_{2}-\bar{\lambda}_{1})\Phi(\xi_{t})dt,$

which is equivalent to

$d\left(e^{(\bar{\lambda}_{2}-\bar{\lambda}_{1})t}\Phi(\xi_{t})\right)\leq\sqrt{2}e^{(\bar{\lambda}_{2}-\bar{\lambda}_{1})t}\Phi^{\prime}(\xi_{t})d\beta_{t}.$

Integrating and taking expectation gives

$\mathbb{E}\left(e^{(\bar{\lambda}_{2}-\bar{\lambda}_{1})t}\Phi(\xi_{t})\right)\leq\Phi(\xi_{0}),$

or equivalently

$\mathbb{E}\Phi(\xi_{t})\leq
e^{-(\bar{\lambda}_{2}-\bar{\lambda}_{1})t}\Phi(\xi_{0}).$ (5.6)

By Lemma 3.2, we know that the above inequality holds for all $t\geq 0$.
Finally, by the definition (5.1) and Itô’s formula, we have
$v_{t}(x)=\mathbb{E}(v_{0}(X_{t}))$. Since $v_{0}$ is Lipschitz on
$\bar{\Omega}$, we can find $K>0$ such that

$\displaystyle\left|v_{t}(x)-v_{t}(y)\right|$
$\displaystyle=\left|\mathbb{E}(v_{0}(X_{t})-v_{0}(Y_{t}))\right|\leq\mathbb{E}\left|v_{0}(X_{t})-v_{0}(Y_{t})\right|$
$\displaystyle\leq K\mathbb{E}\rho(X_{t},Y_{t})=2K\mathbb{E}\xi_{t}.$ (5.7)

Since $\tilde{V}$ is even and (5.5) is the same as equation (2.7) of [18],
Lemma 2.3 of [18] applies to $\Phi$. Thus, $\Phi$ is increasing on $[0,D/2]$
with $\Phi^{\prime}(0)>0$; from which we deduce that there exists $c_{1}>0$,
such that $\Phi(s)\geq c_{1}s$ on $[0,D/2]$. Hence, we have from (5.6), that

$c_{1}\mathbb{E}\xi_{t}\leq\mathbb{E}(\Phi(\xi_{t}))\leq
e^{-(\tilde{\lambda}_{2}-\tilde{\lambda}_{1})t}\Phi(\xi_{0}).$

Putting it together with (5.7) and the definition of $v_{t}$ gives

$e^{-(\lambda_{2}-\lambda_{1})t}\left|v_{0}(x)-v_{0}(y)\right|=\left|v_{t}(x)-v_{t}(y)\right|\leq\frac{2K}{c_{1}}e^{-(\bar{\lambda}_{2}-\bar{\lambda}_{1})t}\Phi(\xi_{0}).$

Since $v_{0}$ is not constant function, we can find $x\neq y$ such that
$\left|v_{0}(x)-v_{0}(y)\right|>0$. Our result follows by sending
$t\rightarrow\infty$. ∎

######  Remark 5.1.

This proof is in the flavor of parabolic proof of Theorem 4.1 in [18], but
simplifies as we do not need to use the preservation of modulus of continuity,
Theorem 4.6 in [18].

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