Spectral gaps for the O-U/Stochastic heat processes on path space over a Riemannian manifold with boundary

TL;DR

This paper extends spectral gap results for Ornstein-Uhlenbeck and stochastic heat processes on path spaces from boundaryless to bounded Riemannian manifolds, broadening understanding of these processes in more general geometric settings.

Contribution

It introduces spectral gap estimates for these processes on manifolds with boundary, a significant generalization of prior boundaryless results.

Findings

Spectral gap estimates for O-U process on manifolds with boundary

Spectral gap estimates for stochastic heat process on manifolds with boundary

Extension of previous results to new geometric setting

Abstract

Fang-Wu\cite{FW17} presented a explicit spectral gap for the O-U process on path space over a Riemannian manifold without boundary under the bounded Ricci curvature conditions. In this paper, we will extend these results to the case of the Riemannian manifold with boundary. Moreover, we also derive the similar results for the stochastic heat process.

Equations

$$K_2\le {\text{Ric}}^Z\le K_1,{\sigma }_2\le \mathbb{I}\le {\sigma }_1$$

$$W_x^T(M)=\{\gamma \in C([0,T];M):{\gamma }_0=x\}.$$

$${\rho }_{\infty }(\gamma ,\sigma ):=\underset{t\in [0,T]}{\sup }\rho ({\gamma }_t,{\sigma }_t),\gamma ,\sigma \in W_x^T(M).$$

$$\begin{array}{r}\text{d}{U}_t^x=H_i(U_t^x)\circ \text{d}{W}_t+H_Z(U_t^x)\text{d}t+H_N(U_t^x)\text{d}{l}_t^x,U_0^x\in O_x(M),\end{array}$$

$$\begin{array}{r}\text{d}{X}_t^x=U_t^x\circ d{W}_t+Z(X_t^x)\text{d}t+N(X_t)\text{d}{l}_t^x,X_0^x=x\end{array}$$

$$\mathbb{H}=\left\{h\in C([0,T];{\mathbb{R}}^d):h(0)=0,\Vert h{\Vert }_{{\mathbb{H}}_{\mathbb{T}}}^2:={\int }_0^T|h_s^{\prime }{|}^2\text{d}s<\infty \right\}.$$

$$Q_{s,t}^{x}=\bigg{(}I-\int_{s}^{t}Q_{s,r}^{x}\big{\{}\text{\rm{Ric}}_{Z}(U_{r}^{x})\text{\rm{d}}s+{\mathbb{I}}(U_{r}^{x})\text{\rm{d}}l_{r}^{x}\big{\}}\bigg{)}\Big{(}I-1_{\{X_{t}^{x}\in\partial M\}}P_{U_{t}^{x}}\Big{)},$$

$$⟨P_{u}a,b⟩:=⟨ua,N⟩⟨ub,N⟩,a,b\in {\mathbb{R}}^d,u\in {\cup }_{x\in \partial M}{O}_x(M).$$

$${\tilde{D}}_{t}F(X_{[0,T]}^x)=\sum _{i:t_i>t}{Q}_{t,t_i}^x{U}_{t_i}^{-1}{\nabla }_{i}f(X_{t_1}^x,\cdots ,X_{t_N}^x),t\in [0,T].$$

$$D_{t}F(X_{[0,T]}^{x})=\sum_{i:t_{i}>t}\Big{(}I-1_{\{X_{t_{i}}^{x}\in\partial M\}}P_{U_{t_{i}}^{x}}\Big{)}U_{t_{i}}^{-1}\nabla_{i}f(X_{t_{1}}^{x},\cdots,X_{t_{N}}^{x}),\ \ t\in[0,T].$$

$${\mu }_t(\text{d}r)$$

$$=:{\phi }_1(t,r,K_1,K_2,{\sigma }_1,{\sigma }_2)\text{d}r+{\phi }_1(t,r,K_1,K_2,{\sigma }_1,{\sigma }_2)\text{d}{l}_r^x$$

$$\displaystyle A_{t}=\Big{(}1+\mu([t,T])\Big{)}+\int_{0}^{t}\Big{(}1+\mu([r,T])\Big{)}\varphi_{1}(r,t,K_{1},K_{2},\sigma_{1},\sigma_{2})\text{\rm{d}}r$$

$$\displaystyle B_{t}=\int_{0}^{t}\Big{(}1+\mu([r,T])\Big{)}\varphi_{2}(r,t,K_{1},K_{2},\sigma_{1},\sigma_{2})\text{\rm{d}}r.$$

$${\mathcal{E}}_{{\sigma }_1,{\sigma }_2}^{K_1,K_2}(F,F):={\int }_{W_x^T(M)}{\int }_0^T|D_{t}F{|}^2(A_t\text{d}t+B_t\text{d}{l}_t)\text{d}{\mathbb{P}}_x.$$

$$\mathbb{E}\bigg{(}F^{2}\log\frac{F^{2}}{\|F\|^{2}_{L^{2}}}\bigg{)}\leq\mathscr{E}_{\sigma_{1},\sigma_{2}}^{K_{1},K_{2}}(F,F),\quad F\in\mathscr{F}C^{\infty}_{T}.$$

$$\mathbb{E}\bigg{(}F^{2}\log\frac{F^{2}}{\|F\|^{2}_{L^{2}}}\bigg{)}\leq C(T,K_{1},K_{2})\int_{W_{x}^{T}(M)}\int^{T}_{0}|D_{t}F|^{2}\text{\rm{d}}t\text{\rm{d}}\mathbb{P}_{x},\quad F\in\mathscr{F}C^{\infty}_{b}(M).$$

$$Spect(\L)^{-1}\leq\frac{1}{2}+\frac{1}{2}\left(1+\frac{|K_{1}|\vee|K_{2}|}{K_{2}}\Big{[}1-\text{\rm{e}}^{-K_{2}T}\Big{]}\right)^{2}.$$

$$SG(\text{L}{)}^{-1}$$

$$\displaystyle\mathbb{E}\bigg{(}F^{2}\log\frac{F^{2}}{\|F\|^{2}_{L^{2}}}\bigg{)}\leq$$

$$+{\int }_{W_x^T(M)}\left({\sigma }_1(1+{\sigma }_1{l}_T^x)T\right){\int }_0^T|D_{t}F{|}^2\text{d}{l}_t\text{d}{\mathbb{P}}_x.$$

$$\displaystyle\mathbb{E}\bigg{(}F^{2}\log\frac{F^{2}}{\|F\|^{2}_{L^{2}}}\bigg{)}\leq$$

$$+{\int }_{W_x^T(M)}\left(2(|{\sigma }_1|\vee |{\sigma }_2|{)}^2\exp \left[(-{\sigma }_2+\epsilon )l_T^x\right]\right){\int }_0^T|D_{t}F{|}^2\text{d}{l}_t\text{d}{\mathbb{P}}_x$$

$$\mathbb{E}\bigg{(}F^{2}\log\frac{F^{2}}{\|F\|^{2}_{L^{2}}}\bigg{)}\leq 2\int_{W^{T}_{x}(M)}\int^{T}_{0}|\tilde{D}_{t}F|^{2}\text{\rm{d}}t\text{\rm{d}}\mathbb{P}_{x}.$$

$${\int }_0^T|{\tilde{D}}_{t}F{|}^2\text{d}t\le {\int }_0^T|D_{t}F{|}^2(A_t\text{d}t+B_t\text{d}{l}_t).$$

$$\Vert {\text{Ric}}^Z\Vert \le |K_1|\vee |K_2|,\Vert \mathbb{I}\Vert \le |{\sigma }_1|\vee |{\sigma }_2|,$$

$$Q_{s,t}^{x}=\bigg{(}I-\int_{s}^{t}Q_{s,r}^{x}\big{\{}\text{\rm{Ric}}^{Z}(U_{r}^{x})\text{\rm{d}}s+{\mathbb{I}}(U_{r}^{x})\text{\rm{d}}l_{r}^{x}\big{\}}\bigg{)}\Big{(}I-1_{\{X_{t}^{x}\in\partial M\}}P_{U_{t}^{x}}\Big{)}.$$

$$\Vert Q_{t,r}^x\Vert \le \exp \left[-K_2(r-t)-{\sigma }_2(l_r^x-l_t^x)\right].$$

$${\tilde{D}}_{t}F(X_{[0,T]}^x)=\sum _{i:t_i>t}{Q}_{t,t_i}^x{U}_{t_i}^{-1}{\nabla }_{i}f(X_{t_1}^x,\cdots ,X_{t_N}^x)=D_{t}F(X_{[0,T]}^x)-$$

$$\displaystyle\sum_{i:t_{i}>t}\int_{t}^{t_{i}}Q_{t,r}^{x}\big{\{}\text{\rm{Ric}}^{Z}(U_{r}^{x})\text{\rm{d}}s+{\mathbb{I}}(U_{r}^{x})\text{\rm{d}}l_{r}^{x}\big{\}}\Big{(}I-1_{\{X_{t}^{x}\in\partial M\}}P_{U_{t_{i}}^{x}}\Big{)}U_{t_{i}}^{-1}\nabla_{i}f(X_{t_{1}}^{x},\cdots,X_{t_{N}}^{x})$$

$$\displaystyle=D_{t}F(X_{[0,T]}^{x})-\int_{t}^{T}Q_{t,r}^{x}\big{\{}\text{\rm{Ric}}^{Z}(U_{r}^{x})D_{r}F(X_{[0,T]}^{x})\text{\rm{d}}r+{\mathbb{I}}(U_{r}^{x})D_{r}F(X_{[0,T]}^{x})\text{\rm{d}}l_{r}^{x}\big{\}}.$$

$$|{\tilde{D}}_{t}F|(X_{[0,T]}^x)\le |D_{t}F|(X_{[0,T]}^x)$$

$$+{\int }_t^T\exp \left[-K_2(r-t)-{\sigma }_2(l_r^x-l_t^x)\right]\{(|K_1|\vee |K_2|)\text{d}r+(|{\sigma }_1|\vee |{\sigma }_2|)\text{d}{l}_r^x\}|D_{r}F|(X_{[0,T]}^x)\}$$

$$=|D_{t}F|(X_{[0,T]}^x)+{\int }_t^T|D_{r}F|{\mu }_t(\text{d}r).$$

Full text

Keywords: Functional inequality; Ricci Curvature; Second fundamental form; Diffusion process; Path space.

1 Introduction

Functional inequality is an important tool to study the spectral gaps for some diffusion operators in the analysis/stochastic analysis field, especially, for the case of infinite dimensional Riemannian path space. For the manifold without boundary, Fang[13] first established the Poincaré inequality for the O-U operator on Riemanian path space by the Clark-Ocean formula, after that the log-Sobolev inequality/(weak)Poincaré inequality have also been established for the O-U Dirichlet form, see e.g. [2, 4, 10, 19, 20, 24, 5, 22, 28, 29, 6] and references therein. Recently Naber[22] gave some characterizations of the uniform bounds of Ricci curvature by the analysis of the path space. Motiviated by this work, Fang and Wu[16] gave the explicit spectral gap of the O-U operator on path space under the Ricci curvature condition that $K_{2}\leq\text{\rm{Ric}}^{Z}(:=\text{\rm{Ric}}+\nabla Z)\leq K_{1}$ and $K_{2}+K_{1}\geq 0$. This condition $K_{2}+K_{1}\geq 0$ is removed by Cheng-Thalimaier[6].

For the manifold with boundary, Wang[27] proved the damped log-Sobolev inequality for the O-U process on path space, but some geometric informations are hidden in this inequality. In this article, our main aim is to present a estimate of the spectral gap for the O-U operator on path space over a manifold (possible with boundary) under the curvature and the second fundamental form conditions

[TABLE]

for some constants $K_{2},K_{1},\sigma_{2},\sigma_{1}\in\mathbb{R}$. In particular, our results cover Fang-Wu’s results and Cheng-Thalmaier’s results. Moreover, we also obtain the estimate of the spectral gap for the stochastic heat process.

To state our main results, we need to introduce some notation. Let $M$ be a $d$-dimensional complete Riemannian manifold possibly with a boundary $\partial M$ and $N$ be the inward unit normal vector field of $\partial M$. Let $L=\frac{1}{2}\Delta+Z$ be the diffusion operator for some $C^{1}$ vector field $Z$, where $\Delta$ is the Laplace operator on $M$.

Denote by the Riemannian path space:

[TABLE]

Let $\rho$ be the Riemannian distance on M. Then $W_{x}^{T}(M)$ is a Polish space under the uniform distance

[TABLE]

Let $O(M)$ be the orthonormal frame bundle over $M$ and $\pi:O(M)\rightarrow M$ be the canonical projection. Furthermore, we choose a canonical orthonormal basis $\{e_{i}\}_{1\leq i\leq d}$ on $\mathbb{R}^{d}$ and a standard orthonormal basis $H_{i}(u):=(H_{ue_{i}})_{1\leq i\leq d}$ for $u\in O(M)$ of horizontal vector fields on $O(M)$. Then the horizontal reflecting diffusion process is the unique solution to the SDEs:

[TABLE]

where $W_{t}$ is the $d$-dimensional Brwonian motion on a complete filtration probability space $(\Omega,\{\mathscr{F}_{t}\}_{t\geq 0},\mathbb{P})$, $H_{Z}$ and $H_{N}$ are the horizontal lift of $Z$ and $N$ respectively, and $l^{x}_{t}$ is an adapted increasing process which increases only when $X_{t}^{x}:=\pi U_{t}^{x}\in\partial M$ which is called the local time of $X_{t}^{x}$ on $\partial M$. Then it is easy to know that $X_{t}^{x}$ solves the equation

[TABLE]

up to the life time $\zeta$( the maximal time of the solution).

Let $\mathscr{F}C^{\infty}_{T}$ be the space of bounded Lipschitz continuous cylinder functions on $W_{x}^{T}(M)$, i.e. for every $F\in\mathscr{F}C^{\infty}_{T}$, there exist some $N\geq 1$ and $0<t_{1}<t_{2}\cdots<t_{N}\leq T$, $f\in C_{Lip}(M^{N})$ such that $F(\gamma)=f(\gamma_{t_{1}},\cdots,\gamma_{t_{N}}),\gamma\in W_{x}^{T}(M)$ , where $C_{Lip}(M^{N})$ is the collection of bounded Lipschitz continuous functions on $M^{N}$. Suppose $\mathbb{H}$ is the standard Cameron-Martin space for $C([0,T];\mathbb{R}^{d})$, i.e.

[TABLE]

In order to construct O-U process on path space by the theory of Dirichlet form, we first introduce the damped Mallavin gradient given by Wang[26]. To do that, we will recommend a multiplicative functional $Q_{s,t}^{x}$, which is first introduced by Hsu [21] to investigate gradient estimate on $P_{t}$. For any fixed $s\geq 0$, $(Q_{s,t}^{x})_{t\geq s}$ is an adapted right-continuous process on $\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ such that $Q_{s,t}^{x}P_{U_{t}^{x}}=0$ if $X_{t}^{x}\in\partial M$ and

[TABLE]

where $P_{u}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is the projection along $u^{-1}N$, i.e.

[TABLE]

For every $F\in\mathscr{F}C_{T}^{\infty}$, by (4.2.1) in [26], the damped gradient is defined by

[TABLE]

Thus, the associated Mallavin gradient will defined as follows:

[TABLE]

For any constants $K_{2},K_{1},\sigma_{2},\sigma_{1}$ with $K_{2}\leq K_{1},\sigma_{2}\leq\sigma_{1}$and each $t\in[0,T]$, let $\mu$ be the random measure on $[0,T]$ given by

[TABLE]

Denote by two measurable functions on $W_{x}^{T}(M)$

[TABLE]

Throughout the article, we assume that the (reflecting if $\partial M$ exists) $L$-diffusion process is non-explosive. Let $\mathbb{P}_{x}$ be the distribution of the $L$-diffusion process $X^{x}$ starting from a fixed point $x$ up to some fixed time $T>0.$ Then $\mathbb{P}_{x}$ is a probability measure on the Riemannian path space $W_{x}^{T}(M).$ Define the following quadratic form by

[TABLE]

The following Logarithmic Sobolev inequality is the main result of this paper.

Theorem 1.1**.**

Assume that $K_{2}\leq\text{\rm{Ric}}^{Z}\leq K_{1}$ and $\sigma_{2}\leq{\mathbb{I}}\leq\sigma_{1}$. Then the following Logarithmic Sobolev inequality holds

[TABLE]

By the above Theorem 1.1, we obtain the following Corollary for two special cases.

Corollary 1.2**.**

$(a)$* Assume that $M$ is a Riemannian manifold without a boundary and $K_{1}\leq\text{\rm{Ric}}^{Z}\leq K_{2}$, then the Logarithmic Sobolev inequality holds*

[TABLE]

In particular, when $K_{2}<0$, we have

[TABLE]

When $K_{2}>0$, we have

[TABLE]

where $\beta=\frac{|K_{1}|\vee|K_{2}|}{K_{2}}$.

$(b)$* Let $M$ be Ricci flat Riemannian manifold with boundary, and we assume that the second fundamental form satisfies $\sigma_{2}\leq{\mathbb{I}}\leq\sigma_{1}$, then*

When $\sigma_{2}\geq 0$, we get that for any $F\in\mathscr{F}C^{\infty}_{T}$

[TABLE]

When $\sigma_{2}<0$, we get that for any $F\in\mathscr{F}C^{\infty}_{T}$

[TABLE]

for some constant $\varepsilon>0$.

Remark 1.3**.**

$(1)$* Wang [26] proved that the damped Logarithmic Sobolev inequality.*

$(2)$* When $M$ is a Riemannian manifold without boundary, and $K_{2}\leq\text{\rm{Ric}}^{Z}\leq K_{1},K_{2}+K_{1}\geq$, Fang-Wu [16] first proved (1.10), later , this result had been extended to the general case of $K_{2}$ and $K_{1}$ by Cheng-Thalimaier[6].*

The rest of this paper is organized as follows: In Section 2, we will prove Theorem 1.1 and Corollary 1.2. The estimate of the spectral gap for the stochastic heat process will be presented in Section 3.

2 Proofs of Theorem 1.1 and Corollary 1.2

2.1 Proof of Theorem 1.1

Proof of Theorem 1.1.

By Theorem 4.4 in [26], we know that the following damped logarithmic Sobolev inequality holds

[TABLE]

Therefore it suffices to show that

[TABLE]

By using the assumptions of $K_{2}\leq\text{\rm{Ric}}^{Z}\leq K_{1}$ and $\sigma_{2}\leq{\mathbb{I}}\leq\sigma_{1}$, we have

[TABLE]

Combining this with (1.14)

[TABLE]

and [27, Theorem 3.2.1], it is easy to derive that

[TABLE]

By the definition of the damped gradient, we get

[TABLE]

Then we have

[TABLE]

The Hölder’s inequality implies that

[TABLE]

Thus, we obtain

[TABLE]

Up to now, we complete the proof. ∎

2.2 Proof of Corollary 1.2

To prove Corollary 1.2, we need some preparations. Let $\beta=\frac{|K_{1}|\vee|K_{2}|}{K_{2}}$ and define

[TABLE]

and

[TABLE]

Similar to the proof of Proposition 3.3 in Fang-Wu[16] for the case of $K_{1}+K_{2}\geq 0$, in the following we will discuss monotonicity of the function $\Lambda(\cdot,T)$.

Proposition 2.1**.**

$(1)$* If $K_{2}<0$, then $t\rightarrow\Lambda(t,T)$ is strictly increasing over $[0,T]$. $(2)$ If $K_{2}\geq 0$, then the maximum is attained at a point $t_{0}$ in $(0,T)$.*

Proof.

According to the definition (2.8) of $\Lambda(t,T)$, we get

[TABLE]

and

[TABLE]

In particular, the second in the above implies that $\Lambda(T,T)\geq\Lambda(0,T)$.

Next, we take the derivative of $\Lambda(t,T)$ with respect to $t$,

[TABLE]

Then we have

[TABLE]

and

[TABLE]

Noting that

[TABLE]

Now we look for $t\in[0,T]$ such that $\Lambda^{\prime}(t,T)=0$. We have

[TABLE]

Therefore there exists at most one $t$ such that $\Lambda^{\prime}(t,T)=0$. For the case where $K_{2}<0$, if there exists $t_{0}\in(0,T)$ such that $\Lambda(t_{0},T)<0$. Then by (2.9) and (2.11), the equation $\Lambda^{\prime}(t,T)=0$ has at least two solutions, it is impossible. Therefore for $K_{2}<0$, $\Lambda^{\prime}(t,T)\geq 0$. For $K_{2}>0$, we suppose $t_{0}$ such that $\Lambda^{\prime}(t_{0},T)=0$, then by (2.12)

[TABLE]

Thus the proof is completed.∎

By the above Proposition 2.1, it is easy to obtain the following Proposition 2.2.

Proposition 2.2**.**

$(i)$* If $K_{2}>0$,*

[TABLE]

$(ii)$* If $K_{2}<0$,*

[TABLE]

Proof of Corollary 1.2.

$(a)$ Since $M$ is a Riemannian manifold without boundary, then the local time $l_{t}=0$, thus we have

[TABLE]

Then

[TABLE]

Which implies that

[TABLE]

Then by (2.15) and the first equality of (2.16),

[TABLE]

Thus we get

[TABLE]

From which we have

[TABLE]

Then (1.9), (1.10) and (1.11) come from Theorem 1.1 and Proposition 2.2.

$(b)$ By the assumption of $M$, we know that

[TABLE]

Thus,

[TABLE]

In addition, (2.20) implies that

[TABLE]

Then, by the definition of $A_{t}$ and $B_{t}$,

[TABLE]

Since $l_{t}^{x}$ is a increasing process, we have

[TABLE]

and

[TABLE]

By Theorem 1.1, when $\sigma_{2}\geq 0$, we get

[TABLE]

and when $\sigma_{2}<0$, we get

[TABLE]

∎

Corollary 2.3**.**

Let $M:=\left\{x=(x_{1},\cdots,x_{d})\in\mathbb{R}^{d}:a_{1}x_{1}+\cdots+a_{d}x_{d}\geq c\right\}$ for some constant $c\in\mathbb{R}$, then $M$ is a Riemannian manifold without boundary and $\text{\rm{Ric}}=0$ with ${\mathbb{I}}=0$, thus we have

[TABLE]

3 Stochastic heat equation

In this section, we will consider the spectral gap for the stochastic heat equation on a Riemannian manifold with boundary. Before moving on, let’s introduce some notation.

The stochastic heat equation on Riemannian manifold had been studied detailed by [23](see also [18]). Here they introduced some notation. In particular, the classical cylinder function depending on finite times is not in the domain of generator associated to the stochastic heat equation. Thus, we need to introduce a class of new cylinder function $\mathscr{F}C_{b}^{1}$ on $W_{x}^{T}(M),$ i.e. for every $F\in\mathscr{F}C_{b}^{1}$, there exist some $m\geq 1,~{}m\in\mathbb{N},~{}f\in C_{b}^{1}(\mathbb{R}^{m}),g_{i}\in C_{b}^{0,1}([0,1]\times M)$, $i=1,...,m$, such that

[TABLE]

where $C_{b}^{0,1}([0,1]\times M)$ denotes the functions which are continuous w.r.t. the first variable and differentiable w.r.t. the second variable with continuous derivatives.

For any $F\in\mathscr{F}C_{b}^{1}$ with (2.2) form and $h\in L^{2}([0,1];\mathbb{R}^{d})$, according to Wang[26], the damped Malliavin gradient of $F$ is given by

[TABLE]

Let $\tilde{\nabla}F$ be the damped $L^{2}$-gradient of $F$, and since

[TABLE]

Then, we have

[TABLE]

Thus,

[TABLE]

The $L^{2}$-gradient of $F$ is defined by

[TABLE]

By Lemma 4.3.2 in [26], we have

[TABLE]

By the standard the procedure, we have

[TABLE]

By (2.3) and Hölder’s inequality, we get

[TABLE]

Let

[TABLE]

Then by changing the order of integration we obtain

[TABLE]

where

[TABLE]

Thus, we get the following Logarithmic Sobolev inequality.

Theorem 3.1**.**

Assume that $\text{\rm{Ric}}^{Z}\geq K$ and ${\mathbb{I}}\geq\sigma$. Then the following Logarithmic Sobolev inequality holds

[TABLE]

Corollary 3.2**.**

$(a)$* Assume that $M$ is a Riemannian manifold with a convex boundary and $K\leq\text{\rm{Ric}}^{Z}$, then the Logarithmic Sobolev inequality holds*

[TABLE]

where

[TABLE]

Proof.

Since the boundary is convex, thus ${\mathbb{I}}\geq\sigma\geq 0$. Thus

[TABLE]

and

[TABLE]

Then we get

[TABLE]

In the following, similar to the argument of Proposition 2.1. Taking the derivative of $s\rightarrow A(s)$ gives

[TABLE]

Thus,

[TABLE]

Noting that

[TABLE]

Now we look for $s\in[0,T]$ such that $A^{\prime}(s)=0$. We have

[TABLE]

Therefore there exists at most one $t$ such that $A^{\prime}(s)=0$. For the case where $K<0$, if there exists $s_{0}\in(0,T)$ such that $A(s_{0})<0$. Then by (3.7) and (3.8), the equation $A^{\prime}(s)=0$ has at least two solutions, it is impossible. Therefore for $K<0$, $A^{\prime}(s)\geq 0$. For $K>0$, we suppose $s_{0}$ such that $\Lambda^{\prime}(t_{0},T)=0$, then by (3.9)

[TABLE]

The proof is completed.

∎

Corollary 3.3**.**

Let $M$ be a Ricci-flat Riemannian manifold with a convex boundary, then we have

[TABLE]