# Perelman’s entropy formula for the Witten Laplacian on manifolds with time
dependent metrics and potentials

Songzi Li, Xiang-Dong Li Research supported by NSFC No. 10971032, Key
Laboratory RCSDS, CAS, No. 2008DP173182, and a Hundred Talents Project of
AMSS, CAS.

Abstract. In this paper, we first give a new proof of the $W$-entropy formula
for the heat equation associated with the Witten Laplacian on compact
Riemannian manifolds with a fixed metric and a fixed potential. This gives a
natural geometric interpretation of a quantity appeared in the $W$-entropy
formula. Then we extend the $W$-entropy formula to the heat equation of the
Witten Laplacian on compact Riemannian manifolds with time dependent metrics
and potentials. As an application, we prove that the optimal logarithmic
Sobolev constant associated with the Witten Laplacian on compact manifolds
equipped with the $m$-dimensional Perelman’s super Ricci flow is decreasing in
time. Finally, we derive the $W$-entropy formula for the backward heat
equation associated with the Witten Laplacian on compact Riemannian manifolds
equipped with Lott’s modified Ricci flow.

##  1 Introduction

Let $M$ be a complete Riemannian manifold with a fixed Riemnnian metric $g$
and a fixed potential $\phi\in C^{2}(M)$. Let $d\mu=e^{-\phi}dv$, where $dv$
is the Riemannian volume measure on $(M,g)$. The Witten Laplacian, called also
the weighted Laplacian,

$\displaystyle L=\Delta-\nabla\phi\cdot\nabla$

is a self-adjoint and non-positive definite operator on $L^{2}(M,\mu)$. By
Itô’s calculus, one can construct the symmetric diffusion process $X_{t}$
associated to the Witten Laplacian by solving the SDE

$\displaystyle dX_{t}=\sqrt{2}dW_{t}-\nabla\phi(X_{t})dt,$

where $W_{t}$ is the Brownian motion on $M$. Moreover, it is well known that
the transition probability density function of the diffusion process $X_{t}$
is exactly the fundamental solution to the heat equation of $L$, i.e., the
heat kernel of the Witten Laplacian $L$. In view of this, it is a fundamental
problem to study the heat equation and the heat kernel of the Witten Laplacian
on manifolds.

In recent years, important progress has been obtained in the study of the heat
equation associated with the Witten Laplacian by using new ideas and new
methods from geometric analysis, PDEs, and probability theory. In particular,
F. Otto [26] introduced an infinite dimensional Riemannian structure on the
Wasserstein space of probability measures on $\mathbb{R}^{n}$ and proved that
the heat equation

$\displaystyle\partial_{t}u=Lu$ (1)

can be realized as the reverse gradient flow of the Boltzmann-Shannon
entropy111Equivalently, the heat equation $(\ref{L-HE})$ is the gradient flow
of $Ent(u)=-H(u)$ on the Wasserstein space $\mathscr{P}_{2}(\mathbb{R}^{n})$
equipped with Otto’s infinite dimensional Riemannian metric.

$H(u)=-\int_{M}u\log ud\mu.$

See also [27, 32, 33, 34, 35] for the extension of Otto’s work to Riemannian
manifolds.

The Witten Laplacian is a natural extension of the standard Laplace Beltrami
operator and has a close connection with differential geometry, probability
theory, quantum field theory and statistical mechanics. In view of this, it is
natural to raise the question whether one can extend the results which hold
for the standard Laplace Beltrami operator to the Witten Laplacian on
manifolds. The main tool which leads such an extension possible is the so-
called Bakry-Emery Ricci curvature associated to $L$ introduced in [2], i.e.,

$\displaystyle Ric(L)=Ric+\nabla^{2}\phi,$

which plays the same rôle as the Ricci curvature for the standard Laplace
Beltrami operator. We refer the reader to [4, 3, 17] for the Li-Yau Harnack
estimates and the heat kernel estimates to the heat equation $(\ref{L-HE})$,
and to [17] for the extension of S.-T. Yau’s Strong Liouville theorem for the
positive $L$-harmonic functions and the $L^{1}$-uniqueness of the heat
equation $(\ref{L-HE})$ on complete Riemannian manifolds. See also [1, 5, 12,
13, 27, 33, 34, 35, 36] for further works on the study of the Witten Laplacian
and Bakry-Emery Ricci curvature on manifolds with weighted measures.

The Bakry-Emery Ricci curvature has been essentially used in Perelman’s work
[30] on the entropy formula for Ricci flow, where Perelman first introduced
the $\mathcal{F}$-energy functional on $\mathcal{M}=\\{{\rm all\ Riemannian\
metrics}\ g\ {\rm on}\ M\\}\times C^{\infty}(M)$ as follows

$\displaystyle\mathcal{F}(g,f)=\int_{M}(R+|\nabla f|^{2})e^{-f}dv,$

where $f\in C^{\infty}(M)$, $R$ denotes the scalar curvature on $(M,g)$, and
$dv$ denotes the volume measure. Under the constraint condition which requires
that

$\displaystyle d\mu=e^{-f}dv$

is a fixed weighted measure on $(M,g)$, Perelman [30] proved that the gradient
flow of the $\mathcal{F}$-energy functional with respect to the standard
$L^{2}$-Riemannian metric on $\mathcal{M}$ is given by the following modified
Ricci flow, which we call Perelman’s Ricci flow in the sequel,

$\displaystyle\partial_{t}g=-2(Ric+\nabla^{2}f),$

and $f$ satisfies the so-called conjugate heat equation

$\displaystyle\partial_{t}f=-\Delta f-R.$

Moreover, Perelman [30] introduced the $W$-entropy functional and proved its
monotonicity for the Ricci flow on compact manifolds. This result plays an
important rôle in the proof of the no local collapsing theorem and in the
final resolution of the Poincaré conjecture and geometrization conjecture (see
e.g. [7, 25, 15]). Since then, many people have derived the $W$-entropy
formula for various geometric evolution equations and used it to study further
analysis and geometric properties of manifolds. See e.g. [10, 8, 15, 28, 29,
11, 23, 24].

In [18], see also [19, 20], inspired by Perelman’s work on the $W$-entropy
formula for Ricci flow, the second author proved the $W$-entropy formula for
the fundamental solution of the Witten-Laplacian on complete Riemannian
manifolds with bounded geometry condition, which extends a previous result due
to Ni [28, 29], who proved an analogue of Perelman’s $W$-entropy formula for
the heat equation $\partial_{t}u=\Delta u$ on complete Riemannian manifolds
with a fixed metric. To be more precise, let

$\displaystyle
Ric_{m,n}(L)=Ric+\nabla^{2}\phi-{\nabla\phi\otimes\nabla\phi\over m-n}$

be the $m$-dimensional Bakry-Emery Ricci curvature associated with the Witten
Laplacian $L$. Then we have the following result.

###### Theorem 1.1

([18, 19, 20]) Let $(M,g)$ be a complete Riemannian manifold with bounded
geometry condition222Here we say that $(M,g)$ satisfies the bounded geometry
condition if the Riemannian curvature tensor ${\rm Riem}$ and its covariant
derivatives $\nabla^{k}{\rm Riem}$ are uniformly bounded on $M$, $k=1,2$., and
$\phi\in C^{4}(M)$ with $\nabla\phi\in C_{b}^{3}(M)$. Let $m>n$, and
$u={e^{-f}\over(4\pi t)^{m/2}}$ be the fundamental associated with the Witten
Laplacian, i.e., the heat kernel to the heat equation $\partial_{t}u=Lu$. Let

$\displaystyle H_{m}(u,t)=-\int_{M}u\log ud\mu-{m\over 2}(1+\log(4\pi t)),$

and define

$\displaystyle W_{m}(u,t):={d\over dt}(tH_{m}(u)).$ (2)

Then

$\displaystyle W_{m}(u,t)=\int_{M}\left[t|\nabla\log
u|^{2}+f-m\right]{e^{-f}\over(4\pi t)^{m/2}}d\mu,$

and

$\displaystyle{dW_{m}(u,t)\over dt}$ $\displaystyle=$
$\displaystyle-2\int_{M}t\left(\left|\nabla^{2}f-{g\over
2t}\right|^{2}+Ric_{m,n}(L)(\nabla f,\nabla f)\right)ud\mu$
$\displaystyle\hskip 56.9055pt-{2\over
m-n}\int_{M}t\left({\nabla\phi\cdot\nabla f}+{m-n\over 2t}\right)^{2}ud\mu.$

In particular, if $(M,g,\phi)$ satisfies the bounded geometry condition and
$Ric_{m,n}(L)\geq 0$, then the $W$-entropy is decreasing in time $t$, i.e.,

$\displaystyle{dW_{m}(u,t)\over dt}\leq 0,\ \ \ \ \ \ \ \forall t\geq 0.$

The purpose of this paper is to extend the $W$-entropy formula in Theorem 1.1
to the heat equation $(\ref{L-HE})$ associated with the time dependent Witten
Laplacian on compact Riemannian manifolds equipped with time dependent metrics
and potentials. In view of Perelman’s work using the $W$-entropy formula for
the Ricci flow to remove “the major stumbling block in Hamilton’s approach to
geometrization” ([30]), it might be possible that the $W$-entropy formula for
the time dependent Witten Laplacian can bring some new insights to the study
of geometric analysis on Riemannian manifolds with time dependent metrics and
potentials.

We now in a position to state the main result of this paper as follows.

###### Theorem 1.2

Let $(M,g(t),t\in[0,T])$ be a family of compact Riemannian manifolds with
potential functions $f(t)\in C^{\infty}(M)$, $t\in[0,T]$. Suppose that $g(t)$
and $f(t))$ satisfies the conjugate equation

$\displaystyle\frac{\partial f}{\partial t}={1\over 2}{\rm
Tr}\left(\frac{\partial g}{\partial t}\right).$

Let

$\displaystyle L=\Delta_{g(t)}-\nabla_{g(t)}f(t)\cdot\nabla_{g(t)}$

be the time dependent Witten Laplacian on $(M,g(t),f(t))$. Let $u$ be a
positive solution of the heat equation

$\displaystyle\partial_{t}u=Lu$

with initial date $u(0)$ satisfying $\int_{M}u(0)d\mu(0)=1$. Let

$\displaystyle H_{m}(u,t)=-\int_{M}u\log ud\mu-{m\over 2}(1+\log(4\pi t)),$

and define

$\displaystyle W_{m}(u,t):={d\over dt}(tH_{m}(u)).$

Then

$\displaystyle W_{m}(u,t)=\int_{M}\left[t|\nabla\log u|^{2}-\log
u-m\right]ud\mu,$

and

$\displaystyle{d\over dt}W_{m}(u,t)$ $\displaystyle=$
$\displaystyle-2t\int_{M}\left|\nabla^{2}\log u+{g\over
2t}\right|^{2}ud\mu-{2t\over m-n}\int_{M}\left(\nabla f\cdot\nabla\log
u-{m-n\over 2t}\right)^{2}ud\mu$ $\displaystyle\ \ \ \ \ \ \ \ \
-2\int_{M}t\left({1\over 2}{\partial g\over\partial
t}+Ric_{m,n}(L)\right)(\nabla\log u,\nabla\log u)ud\mu.$

In particular, if $\\{g(t),f(t),t\in(0,T]\\}$ satisfies the $m$-dimensional
Perelman’s super Ricci flow and the conjugate equation

$\displaystyle{1\over 2}{\partial g\over\partial t}+Ric_{m,n}(L)\geq 0,$ (3)
$\displaystyle\hskip 14.22636pt\frac{\partial f}{\partial t}={1\over 2}{\rm
Tr}\left(\frac{\partial g}{\partial t}\right),$ (4)

then $W_{m}(u,t)$ is decreasing in $t\in(0,T]$, i.e.,

$\displaystyle{dW_{m}(u,t)\over dt}\leq 0,\ \ \ \forall t\in(0,T].$

As an application of the $W$-entropy formula for the Witten Laplacian on
manifolds with time dependent metrics and potentials, we prove that the
optimal logarithmic Sobolev constant associated with the Witten Laplacian on
compact manifolds equipped with the $m$-dimensional Perelman’s super Ricci
flow is decreasing in time. More precisely, we have

###### Theorem 1.3

Let $(M,g(t),f(t),t\in[0,T])$ be as in Theorem 1.2. Then there exists a
positive and smooth function $u=e^{-{v\over 2}}$ such that $v$ achieves the
optimal logarithmic Sobolev constant $\mu(t)$ defined by

$\displaystyle\mu(t):=\inf\limits\left\\{W_{m}(u,t):\
\int_{M}{e^{-v}\over(4\pi t)^{m/2}}d\mu=1\right\\}.$

Indeed, $u=e^{-{v\over 2}}$ is a solution to the nonliniear PDE

$\displaystyle-4tLu-2u\log u-mu=\mu(t)u.$

Moreover, if $\\{g(t),f(t),t\in[0,T]\\}$ satisfies the $m$-dimensional
Perelman’s super Ricci flow $(\ref{Cond-1})$ and the conjugate equation
$(\ref{Cond-2})$, then $\mu(t)$ is decreasing in $t$ on $[0,T]$.

###### Remark 1.4

In [16], Li and Yau proved the Li-Yau Harnack inequality

$\displaystyle\Delta\log u+{n\over 2t}\geq 0$

for any positive solution to the heat equation $\partial_{t}u=\Delta u$ on
complete Riemannian manifolds with non-negative Ricci curvature. In general,
it was proved in [17, 18] that for any positive solution to the heat equation
$\partial_{t}u=Lu$, the Li-Yau Harnack inequality

$\displaystyle L\log u+{m\over 2t}\geq 0$

holds on complete Riemannian manifolds with $Ric_{m,n}(L)\geq 0$. See also [4,
3]. By [18], we have the following entropy formula

$\displaystyle{d\over dt}H_{m}(u,t)=-\int_{M}\left(L\log u+{m\over
2t}\right)ud\mu.$ (5)

Therefore, $H_{m}(u,t)$ is decreasing in time $t$ provided that
$Ric_{m,n}(L)\geq 0$, i.e.,

$\displaystyle{d\over dt}H_{m}(u,t)\leq 0.$

###### Remark 1.5

In [30], Perelman gave an interpretation of the $W$-entropy using the
Boltzmann entropy formula in statistical mechanics. In [18, 19], the second
author gave a probabilistic interpretation of the $W$-entropy for the Ricci
flow, the heat equation of the Witten Laplacian and for the Fokker-Planck heat
equation. Note that, as noticed in [18, 19, 20],

$\displaystyle H_{m}(u,t)=H(u,t)-H(\gamma,t)$

where $H(u,t)$ is the Boltzmann-Shannon entropy associated with the heat
equation to the Witten Laplacian on $(M,g(t),f(t))$, and $H(\gamma,t)$ is the
Boltzmann-Shannon entropy of the Gaussian heat kernel $\gamma(x,t)$ on
$\mathbb{R}^{m}$ if $m\in\mathbb{N}$ with $m\geq n$, i.e.,

$\displaystyle\gamma(x,t)={1\over(4\pi t)^{m/2}}e^{-{|x|^{2}\over 4t}},\ \ \
x\in\mathbb{R}^{m},t>0.$

In view of the definition formula $(\ref{BPF})$, the $W$-entropy $W_{m}(u,t)$
can be regarded as the by-product of the Boltzmann-Shannon entropy. This gives
a probabilistic interpretation of the $W$-entropy $W_{m}(u,t)$.

On the other hand, similarly to Perelman [30], we can also give a heuristical
interpretation of the $W$-entropy using the Boltzmann entropy formula in
statistical mechanics: Suppose that there exists a canonical ensemble with a
“density of state measure” $g(E)dE$ such that the partition function
$Z_{\beta}=\int_{\mathbb{R}^{+}}e^{-\beta E}g(E)dE$ is given by

$\displaystyle\log Z_{\beta}=H_{m}(u,t),$ (6)

where $t=\beta^{-1}$. Here, as in [30], we do not discuss the issue whether
such a “density of state measure” exists or not. Then, formally applying the
Boltzmann entropy formula in statistical mechanics, the thermodynamical
entropy of this canonical ensemble is given by

$S=\log Z_{\beta}-\beta{\partial\over\partial\beta}\log Z_{\beta}.$

Using the fact ${\partial\over\partial\beta}={\partial\over\partial
t}{\partial t\over\partial\beta}=-{1\over\beta^{2}}{\partial\over\partial
t}=-t^{2}{\partial\over\partial t}$, we have

$\displaystyle S=H_{m}(u,t)+t{\partial\over\partial t}H_{m}(u,t).$

Thus

$S=W_{m}(u,t).$

Moreover, as noticed in [30], formally using the formula in statistical
mechanics

$\displaystyle{dS\over d\beta}=-\beta{\partial^{2}\over\partial\beta^{2}}\log
Z_{\beta},$

we can reprove the $W$-entropy formula in Theorem 1.2.

###### Remark 1.6

Developing the approach used in [18, 19, 20], it would be possible to further
extend the $W$-entropy formula in Theorem 1.2 to the fundamental solution of
the heat equation associated with the Witten Laplacian on complete Riemannian
manifolds with time dependent metrics and potentials satisfying the bounded
geometry condition. Technically, this requires to prove some Hamilton type
gradient estimates (see [14]) for the logarithmic of the heat kernel of the
Witten Laplacian on complete Riemannian manifolds with time dependent metrics
and potentials satisfying the uniformly bounded geometry condition333Here we
say that $(M,g(t),f(t),t\in[0,T])$ satisfies the uniformly bounded geometry
condition if there exists some $N\in\mathbb{N}$ such that for all
$\varepsilon\in(0,T)$, the $k$-th order covariant derivatives of the
Riemannian curvature tensor ${\rm Riem(g(t))}$, i.e., $\nabla^{k}{\rm
Riem(g(t))}$, as well as the $k$-th order covariant derivatives of $f(t)$,
i.e., $\nabla^{k}f(t)$ are uniformly bounded on $[\varepsilon,T]\times M$,
$k=0,...,N$.. We will study this problem in a forthcoming paper. If this can
be verified, we can derive that, if $\\{g(t),f(t),t\in(0,T]\\}$ is a family of
metrics and potentials satisfying $(\ref{Cond-1})$ and $(\ref{Cond-2})$ on a
complete Riemanian manifold $M$ with uniformly bounded geometry condition,
then

$\displaystyle{d\over dt}W_{m}(u,t)=0,\ \ \ \ \ \ {\rm for\ \ some}\ \
t=\tau\in(0,T],$

if and only if

$\displaystyle\nabla^{2}\log u$ $\displaystyle=$ $\displaystyle-{g\over
2\tau},$ $\displaystyle\nabla f\cdot\nabla\log u$ $\displaystyle=$
$\displaystyle{m-n\over 2\tau}.$

By the same argument as used in [18, 19, 20], we can further prove the
following rigidity theorem: Let $\\{g(t),f(t),t\in(0,T]\\}$ be a family of
metrics and potentials satisfying $(\ref{Cond-1})$ and $(\ref{Cond-2})$ on a
complete Riemanian manifold $M$ with uniformly bounded geometry condition. Let
$u$ be the fundamental solution to the heat equation $\partial_{t}u=Lu$. Then,

$\displaystyle{d\over dt}W_{m}(u,t)=0,\ \ \ \ \ \ {\rm for\ \ some}\ \
t=\tau\in(0,T],$

if and only if $(M,g(t))$ is isometric to $\mathbb{R}^{n}$, $f(t)$ is
identically equal to a constant, $m=n$, and

$u(x,t)={e^{-{\|x\|^{2}\over 4t}}\over(4\pi t)^{n/2}},\ \ \ \forall\ x\in
M=\mathbb{R}^{n},\ \ t\in(0,T].$

The rest of this paper is organized as follows. In Section $2$, we first give
a new proof444 One of the advantages of our new proof is that it gives a
natural geometric interpretation of the third term appeared in the $W$-entropy
formula $(\ref{W-1})$. See Remark 2.2. to Theorem 1.1. In Section $3$, we
prove the dissipation formula of the Boltzmann-Shannon entropy for the heat
equation of the Witten Laplacian on compact manifolds with time dependent
metrics and potentials. In Section $4$, we prove Theorem 1.2 and Theorem 1.3.
In Section $5$, we use Perelman’s $W$-entropy formula for Ricci flow to derive
the $W$-entropy formula for the backward heat equation of the Witten Laplacian
on compact Riemannian manifolds equipped with a modified Ricci flow introduced
by Lott [22].

##  2 A new proof of Theorem 1.1

To give a new proof of Theorem 1.1, we first recall some elementary geometric
formulas on warped product metrics.

Let $m\in\mathbb{N}$, $m\geq n$. Let $\widetilde{M}=M\times N$, where
$(N,g_{N})$ is a compact Riemannian manifold with dimension $q=m-n$. Let
$\phi\in C^{2}(M)$. We consider the following warped product metric on
$\widetilde{M}$:

$\displaystyle\widetilde{g}=g_{M}\bigoplus e^{-{2\phi\over q}}g_{N}.$ (7)

Let $\nu_{N}$ be the volume measure on $N$. Then the volume measure on
$(\widetilde{M},\widetilde{g})$ is given by

$dvol_{\widetilde{M}}=e^{-\phi}dvol_{M}\otimes d\nu_{N}.$

Denote

$d\mu=e^{-\phi}dvol_{M}.$

Then

$dvol_{\widetilde{M}}=d\mu\otimes d\nu_{N}.$

Without loss of generality, we may assume that

$\nu_{N}(N)=1.$

Let $\widetilde{\Gamma}$ be the Christoffel symbol on
$(\widetilde{M},\widetilde{g})$. By direct calculation, we can verify that

$\displaystyle\widetilde{\Gamma}_{ij}^{k}=\Gamma_{ij}^{k},\ \ \
\widetilde{\Gamma}_{\alpha\beta}^{k}=q^{-1}g^{kl}\partial_{l}\phi
g_{\alpha\beta},\ \ \
\widetilde{\Gamma}_{\alpha\beta}^{\gamma}=\Gamma_{\alpha\beta}^{\gamma},$

and

$\displaystyle\widetilde{\Gamma}_{ij}^{\alpha}=0,\ \
\widetilde{\Gamma}_{i\alpha}^{k}=0,\ \
\widetilde{\Gamma}_{i\alpha}^{\beta}=0.$

Let $\widetilde{\nabla}$ be the Levi-Civita connection on
$(\widetilde{M},\widetilde{g})$. For any $f\in C^{2}(M)$, using the formula

$\displaystyle\widetilde{\nabla}_{ij}^{2}f$ $\displaystyle=$
$\displaystyle\partial_{i}\partial_{j}f-\widetilde{\Gamma}_{ij}^{k}\partial_{k}f,$
$\displaystyle\widetilde{\nabla}_{i\alpha}^{2}f$ $\displaystyle=$
$\displaystyle\partial_{i}\partial_{\alpha}f-\widetilde{\Gamma}_{i\alpha}^{k}\partial_{k}f,$
$\displaystyle\widetilde{\nabla}_{\alpha\beta}^{2}f$ $\displaystyle=$
$\displaystyle\partial_{\alpha}\partial_{\beta}f-\widetilde{\Gamma}_{\alpha\beta}^{k}\partial_{k}f,$

we have

$\displaystyle\widetilde{\nabla}_{ij}^{2}f$ $\displaystyle=$
$\displaystyle\nabla_{ij}^{2}f,$ (8)
$\displaystyle\widetilde{\nabla}_{\alpha\beta}^{2}f$ $\displaystyle=$
$\displaystyle-q^{-1}g^{kl}\partial_{l}\phi\partial_{k}fg_{\alpha\beta},$ (9)
$\displaystyle\nabla_{i\alpha}^{2}f$ $\displaystyle=$ $\displaystyle 0.$ (10)

Hence

$\displaystyle\left|\widetilde{\nabla}^{2}f-{\widetilde{g}\over
2t}\right|^{2}$ $\displaystyle=$ $\displaystyle\left|\nabla^{2}f-{g\over
2t}\right|^{2}+\left|\widetilde{\nabla}_{\alpha\beta}^{2}f-{g_{\alpha\beta}\over
2t}\right|^{2}$ (11) $\displaystyle=$ $\displaystyle\left|\nabla^{2}f-{g\over
2t}\right|^{2}+\left|{g^{kl}\partial_{l}\phi\partial_{k}fg_{\alpha\beta}\over
q}+{g_{\alpha\beta}\over 2t}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|\nabla^{2}f-{g\over
2t}\right|^{2}+\left|\left({\nabla\phi\cdot\nabla f\over m-n}+{1\over
2t}\right)g_{\alpha\beta}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|\nabla^{2}f-{g\over 2t}\right|^{2}+{1\over
m-n}\left({\nabla\phi\cdot\nabla f}+{m-n\over 2t}\right)^{2}.$

The following result was obtained in a private discussion by Bing-Long Chen
and the second author in January 2006.

###### Theorem 2.1

([9]) The Laplace-Beltrami operator on $(\widetilde{M},\widetilde{g})$ is
given by

$\displaystyle\Delta_{\widetilde{M}}=L+e^{-{2\phi\over m-n}}\Delta_{N}.$

Proof. The proof can be given by a direct calculation. $\square$

A new proof of Theorem 1.1. To avoid technical issue, we only consider the
case of compact manifolds. Let $u={e^{-f}\over(4\pi
t)^{m/2}}:M\rightarrow[0,\infty)$ be a positive solution to the heat equation
$\partial_{t}u=Lu$. Then it satisfies the following heat equation on
$(\widetilde{M},\widetilde{g})$

$\displaystyle\partial_{t}u=\Delta_{\widetilde{M}}u.$

Since $f$ depends only on the variable in the $M$-direction, we have
$\widetilde{\nabla}f=\nabla f$. Therefore the $W$-entropy functional
$W_{m}(u,t)$ defined by $(\ref{W-0})$ coincides with the $W$-entropy
functional $\widetilde{W}_{m}(u,t)$ defined on $(\widetilde{M},\widetilde{g})$
as follows

$\displaystyle\widetilde{W}_{m}(u,t)=\int_{\widetilde{M}}\left[t|\widetilde{\nabla}f|^{2}+f-m\right]{e^{-f}\over(4\pi
t)^{m/2}}dvol_{\widetilde{M}}.$ (12)

Applying the $W$-entropy formula for the heat equation $\partial_{t}u=\Delta
u$ on compact Riemannian manifolds with fixed metric due to Ni [28, 29] to
$(\widetilde{M},\widetilde{g})$, we have

$\displaystyle{d\widetilde{W}_{m}(u,t)\over dt}$ $\displaystyle=$
$\displaystyle-2\int_{\widetilde{M}}t\left(\left|\widetilde{\nabla}^{2}f-{\widetilde{g}\over
2t}\right|^{2}+\widetilde{Ric}(\widetilde{\nabla}\log u,\widetilde{\nabla}\log
u)\right)ud\mu dv_{N}.$ (13)

By $(\ref{ccc})$, we have

$\displaystyle\left|\widetilde{\nabla}^{2}f-{\widetilde{g}\over
2t}\right|^{2}=\left|\nabla^{2}f-{g\over 2t}\right|^{2}+{2\over
m-n}\left({\nabla\phi\cdot\nabla f}+{m-n\over 2t}\right)^{2}.$ (14)

On the other hand, by [6, 21, 17], we have

$\displaystyle\widetilde{Ric}(\widetilde{\nabla}\log u,\widetilde{\nabla}\log
u)=Ric_{m,n}(L)(\nabla\log u,\nabla\log u).$ (15)

From $(\ref{WP-W-1})$, $(\ref{WP-Hess})$ and $(\ref{Ric-WP})$, we obtain
$(\ref{W-1})$. This finishes the new proof of Theorem 1.1. $\square$

###### Remark 2.2

One of the advantages of the above proof is that: when $m\in\mathbb{N}$ and
$m>n$, the quantity ${1\over{m-n}}\left(\nabla\phi\cdot\nabla f+{m-n\over
2t}\right)^{2}$ appeared in the $W$-entropy formula in Theorem 1.1 has a
natural geometric interpretation. It corresponds to the vertical component of
the quantity $\left|\widetilde{\nabla}^{2}f-{\widetilde{g}\over
2t}\right|^{2}$ on the warped product manifold $\mathcal{M}=M\times N$
equipped with the metric

$\displaystyle\widetilde{g}=g\bigoplus e^{-{2\phi\over m-n}}g_{N}.$

##  3 Dissipation formula of the Boltzmann-Shannon entropy

Let $(M,g(t),f(t))$ be as in Theorem 1.2. Following [2, 21, 17], we introduce
the Bakry-Emery Ricci curvature associated with $L$ as

$Ric(L)=Ric+\nabla^{2}f.$

The purpose of this section is to prove the following dissipation formula for
the Boltzmann-Shannnon entropy associated with the Witten Laplacian on
manifolds with time dependent metrics and potentials.

###### Theorem 3.1

Let $u$ be a positive solution to the heat equation $\partial_{t}u=Lu$. Let

$H(u,t)=-\int_{M}u\log ud\mu$

be the Boltzmann-Shannon entropy associated with the Witten Laplacian $L$.
Then

$\displaystyle{\partial^{2}\over\partial
t^{2}}H(u,t)=-2\int_{M}\left[|\nabla^{2}\log u|^{2}+\left({1\over 2}{\partial
g\over\partial t}+Ric(L)\right)(\nabla\log u,\nabla\log u)\right]ud\mu.$ (16)

Proof. By direct calculation, we have

$\displaystyle{\partial\over\partial t}H(u,t)=-\int_{M}\partial_{t}u(\log
u+1)d\mu=-\int_{M}Lu(\log u+1)d\mu.$

Integrating by parts yields

$\displaystyle{\partial\over\partial t}H(u,t)=\int_{M}|\nabla\log
u|^{2}_{g(t)}ud\mu,$

which further implies that, as $\partial_{t}(d\mu)=0$, we have

$\displaystyle{\partial^{2}\over\partial t^{2}}H(u,t)$ $\displaystyle=$
$\displaystyle\int_{M}{\partial\over\partial t}(|\nabla\log
u|^{2}_{g(t)}u)d\mu$ (17) $\displaystyle=$
$\displaystyle\int_{M}\left[{\partial\over\partial t}g^{ij}\nabla_{i}\log
u\nabla_{j}\log u\right]ud\mu+\int_{M}{\partial\over\partial t}\left[{|\nabla
u|^{2}\over u}\right]_{\rm g(t)\ fixed}d\mu$ $\displaystyle=$
$\displaystyle\int_{M}\left[-{\partial\over\partial t}g_{ij}\nabla_{i}\log
u\nabla_{j}\log u\right]ud\mu+\int_{M}{\partial\over\partial t}\left[{|\nabla
u|^{2}\over u}\right]_{\rm g(t)\ fixed}d\mu$ $\displaystyle=$
$\displaystyle\int_{M}\left(-{\partial g\over\partial t}(\nabla\log u,\nabla
u)+{\partial\over\partial t}\left[{|\nabla u|^{2}\over u}\right]_{\rm g(t)\
fixed}\right)d\mu,$

where $[\cdot]_{\rm g(t)\ fixed}$ means that the quantity $|\nabla u|^{2}$ in
$[\cdot]$ is defined under a fixed metric $g(t)$. and we have used the facts
$|\nabla\log u|^{2}=g^{ij}\nabla_{i}\log u\nabla_{j}\log u$ as well as
$\partial_{t}g^{ij}=-\partial_{t}g_{ij}$.

By the entropy dissipation formula in [2, 20], we have

$\displaystyle\int_{M}{\partial\over\partial t}\left[{|\nabla u|^{2}\over
u}\right]_{\rm g(t)\ fixed}d\mu=-2\int_{M}\left[|\nabla^{2}\log
u|^{2}+Ric(L)(\nabla\log u,\nabla\log u)\right]ud\mu.$ (18)

Combining $(\ref{aaa})$ and $(\ref{bbb})$, we finish the proof of Theorem 3.1.
$\square$

As an easy consequence of Theorem 1.2, we have

###### Corollary 3.2

Let $(M,g(t))$ be a closed manifold with a potential $f(t)$. Suppose that
$(g(t),f(t))$ satisfies Perelman’s super Ricci flow and the conjugate
equation, i.e.,

$\displaystyle{\partial g\over\partial t}$ $\displaystyle\geq$
$\displaystyle-2Ric(L),$ $\displaystyle\frac{\partial f}{\partial t}$
$\displaystyle=$ $\displaystyle{1\over 2}{\rm Tr}\left(\frac{\partial
g}{\partial t}\right).$

Let $u$ be a positive solution to the heat equation $\partial_{t}u=Lu$. Then
the Boltzmann-Shannon entropy

$H(u,t)=-\int_{M}u\log ud\mu$

is concave in time $t$, i.e.,

$\displaystyle{d^{2}\over dt^{2}}H(u,t)\leq 0.$

##  4 Proofs of Theorem 1.2 and Theorem 1.3

Following [20], we introduce

$\displaystyle W(u,t)={d\over dt}(tH(u,t)).$

By direct calculation, we can prove the following

###### Proposition 4.1

We have

$\displaystyle W(u,t)=\int_{M}\left[t|\nabla\log u|^{2}-\log u\right]ud\mu,$

and

$\displaystyle{d\over dt}W(u,t)$ $\displaystyle=$
$\displaystyle-2\int_{M}t\left[|\nabla^{2}\log u|^{2}+\left({1\over
2}{\partial g\over\partial t}+Ric(L)\right)(\nabla\log u,\nabla\log
u)\right]ud\mu$ (19) $\displaystyle\hskip 85.35826pt+2\int_{M}|\nabla\log
u|^{2}ud\mu.$

###### Remark 4.2

From $(\ref{www})$, we can derive that, if

${1\over 2}{\partial g\over\partial t}+Ric(L)-{1\over t}\geq 0.$

Then

${dW(u,t)\over dt}\leq 0.$

Let

$\displaystyle H_{m}(u,t)=-\int_{M}u\log ud\mu-{m\over 2}(1+\log(4\pi t)).$

Following [30, 28, 18, 20], we define $W_{m}(u,t)$ by the Boltzmann entropy
formula

$\displaystyle W_{m}(u,t)={d\over dt}(tH_{m}(u)).$ (20)

We can verify that $W_{m}(u,t)$ coincides with the expression given in Theorem
1.2, i.e.,

$\displaystyle W_{m}(u,t)=\int_{M}\left[t|\nabla\log u|^{2}-\log
u\right]ud\mu-{m\over 2}(2+\log(4\pi t)).$

Proof of Theorem 1.2. By $(\ref{BPF})$ and $(\ref{2ndH-3})$ in Theorem 3.1, we
have

$\displaystyle{dW_{m}(u)\over dt}$ $\displaystyle=$
$\displaystyle-2\int_{M}t\left[|\nabla^{2}\log u|^{2}+\left({1\over
2}{\partial g\over\partial t}+Ric(L)\right)(\nabla\log u,\nabla\log
u)\right]ud\mu$ (21) $\displaystyle\ \ \ \ \ \ \ +2\int_{M}|\nabla\log
u|^{2}ud\mu-{m\over 2t}.$

Note that

$\displaystyle 2t|\nabla^{2}\log u|^{2}+{m\over 2t}=2t\left|\nabla^{2}\log
u+{g\over 2t}\right|^{2}+{m-n\over 2t}-2\Delta\log u.$

Hence

$\displaystyle{d\over dt}W_{m}(u,t)$ $\displaystyle=$ $\displaystyle-{m-n\over
2t}-2t\int_{M}\left|\nabla^{2}\log u+{g\over
2t}\right|^{2}ud\mu+2\int_{M}|\nabla\log u|^{2}ud\mu+2\int_{M}\Delta\log
uud\mu$ $\displaystyle-2\int_{M}t\left({1\over 2}{\partial g\over\partial
t}+Ric(L)\right)(\nabla\log u,\nabla\log u)ud\mu.$

Integrating by part yields

$\displaystyle\int_{M}\Delta\log uud\mu$ $\displaystyle=$
$\displaystyle\int_{M}(L\log u+\nabla f\cdot\nabla\log u)ud\mu$
$\displaystyle=$ $\displaystyle-\int_{M}|\nabla\log u|^{2}ud\mu+\int_{M}\nabla
f\cdot\nabla\log uud\mu,$

whence

$\displaystyle{d\over dt}W_{m}(u,t)$ $\displaystyle=$ $\displaystyle-{m-n\over
2t}-2t\int_{M}\left|\nabla^{2}\log u+{g\over
2t}\right|^{2}ud\mu+2\int_{M}\nabla f\cdot\nabla\log uud\mu$
$\displaystyle-2\int_{M}t\left({1\over 2}{\partial g\over\partial
t}+Ric(L)\right)(\nabla\log u,\nabla\log u)ud\mu.$

Note that

$\displaystyle{m-n\over 2t}+2tRic(L)(\nabla\log u,\nabla\log u)-2\nabla
f\cdot\nabla\log u$ $\displaystyle=$ $\displaystyle 2tRic_{m,n}(L)(\nabla\log
u,\nabla\log u)+{2t\over m-n}\left(\nabla f\cdot\nabla\log u-{m-n\over
2t}\right)^{2}.$

Hence

$\displaystyle{d\over dt}W_{m}(u,t)$ $\displaystyle=$
$\displaystyle-2t\int_{M}\left|\nabla^{2}\log u+{g\over
2t}\right|^{2}ud\mu-{2t\over m-n}\int_{M}\left(\nabla f\cdot\nabla\log
u-{m-n\over 2t}\right)^{2}ud\mu$ $\displaystyle-2\int_{M}t\left({1\over
2}{\partial g\over\partial t}+Ric_{m,n}(L)\right)(\nabla\log u,\nabla\log
u)ud\mu.$

This proves the $W$-entropy formula in Theorem 1.2. The monotonicity result
follows. The proof of Theorem 1.2 is completed. $\square$

Proof of Theorem 1.3. The proof is similar to Perelman’s monotonicity theorem
for the $\mu$-invariant for Ricci flow [30]. See also [18]. By definition, we
have

$\displaystyle\mu(t)=\inf\limits_{u}\left\\{\int_{M}\left[4t|\nabla
u|^{2}-u^{2}\log u^{2}-mu^{2}\right](4\pi t)^{-{m\over 2}}d\mu\right\\},$ (22)

where $\inf\limits$ is taken among all the $u$ such that

$\int_{M}(4\pi t)^{-{m\over 2}}u^{2}d\mu=1.$

Indeed, $\mu(t)$ is the optimal constant in the following logarithmic Sobolev
inequality: for all $u$ satisfying the above condition,

$\displaystyle\int_{M}u^{2}\log u^{2}(4\pi t)^{-{m\over
2}}d\mu\leq-\mu(t)-m+4\int_{M}t|\nabla u|^{2}(4\pi t)^{-{m\over 2}}d\mu.$

By a similar argument as used in Perelman [30] and [7, 15, 25], we can prove
that the minimization problem $(\ref{optmu})$ has a non-negative minimizer
$u\in H^{1}(M,\mu)$, which satisfies the Euler-Lagrange equation

$\displaystyle-4tLu-2u\log u-mu=\mu(t)u.$

By the regularity theory of elliptic PDEs, we have $u\in C^{1,\alpha}(M)$. By
an argument due to Rothaus [31], we can further prove that $u$ is strictly
positive and smooth. Hence $v=-2\log u$ is also smooth. Moreover, as a
consequence of Theorem 1.2, we can derive that $\mu(t)$ is a decreasing
function in $t$ on $[0,T]$ provided that $\\{g(t),f(t),t\in[0,T]\\}$ satisfies
the $m$-dimensional Perelman’s super Ricci flow $(\ref{Cond-1})$ and the
conjugate equation $(\ref{Cond-2})$. The proof of Theorem 1.3 is completed.
$\square$

###### Remark 4.3

Let $m\in\mathbb{N}$ and $m>n$. Let $(N,g_{N})$ be a compact Riemniann
manifold of dimension $q=m-n$. Let $\mathcal{M}=M\times N$ be the product
manifold equipped with the time dependent warped product metric

$\displaystyle\widetilde{g}(t)=g(t)\bigoplus e^{-{2f(t)\over m-n}}g_{N}.$

Similarly to Remark 2.2, the quantity ${1\over{m-n}}\left(\nabla
f\cdot\nabla\log u-{m-n\over 2t}\right)^{2}$ appeared in the $W$-entropy
formula in Theorem 1.2 has a natural geometric interpretation. It corresponds
to the vertical component of the quantity $\left|\widetilde{\nabla}^{2}\log
u+{\widetilde{g}\over 2t}\right|^{2}$ on $(\mathcal{M},\widetilde{g}(t))$.

##  5 The $W$-entropy for the Ricci flow on warped product manifolds

Let $m\in\mathbb{N}$ and $m\geq n$. Let $\mathbb{T}^{q}$ be the
$q$-dimensional torus with a fixed flat metric given in local coordinates by
$\sum^{q}_{i=1}dx^{2}_{i}$, where $q=m-n$. Let
$\widetilde{M}=M\times\mathbb{T}^{q}$ be equipped with a time dependent warped
product metric

$\displaystyle\widetilde{g}(t)=\sum^{n}_{i,j=1}g_{ij}(t)dx^{i}dx^{j}+u(t)^{\frac{2}{q}}\sum^{q}_{\alpha=1}dx^{2}_{\alpha}.$

In [22], Lott studied the Ricci flow $\widetilde{g}(t)$ on the warped product
manifold $\widetilde{M}=M\times\mathbb{T}^{q}$, which consists of a modified
Ricci flow for the Riemannian metric $g(t)$ and a forward heat equation for a
potential function $\psi(t)=-\log u(t)$ on the manifold $M$. In this section
we use the Perelman’s $W$-entropy formula for the Ricci flow
$\widetilde{g}(t)$ on the warped product manifold $\widetilde{M}$ to derive
the $W$-entropy formula for the backward heat equation associated with the
Witten Laplacian $L=\Delta_{g(t)}-\nabla_{g(t)}\psi(t)\cdot\nabla_{g(t)}$ on
the compact manifold $M$ equipped with Lott’s modified Ricci flow $g(t)$ and
the time dependent potential $\psi(t)$.

We first recall Lott’s Ricci flow on $\widetilde{M}=M\times\mathbb{T}^{q}$.
Let $u=e^{-\psi}$. Let $Ric_{q}$ be the Ricci curvature on
$(\widetilde{M},\widetilde{g})$, and $Ric$ the Ricci curvature on $(M,g)$. By
calculation on warped product manifolds, see [6, 21, 22], we have

$\displaystyle
Ric_{q}=Ric^{q}_{\psi}+\frac{1}{q}(\Delta\psi-|\nabla\psi|^{2})u^{\frac{2}{q}}\sum^{q}_{\alpha=1}dx^{2}_{\alpha},$
(23)

where $Ric^{q}_{\psi}$ is the $m$-dimensional Bakry-Emery Ricci curvature on
$(M,g)$ with respect to the potential function $\psi$, i.e.,

$Ric^{q}_{\psi}:=Ric+Hess\psi-\frac{1}{q}\nabla\psi\otimes\nabla\psi.$

See [2, 18, 19, 20]. By $(\ref{PPP1})$, the scalar curvature on
$(\widetilde{M},\widetilde{g})$, denoted by $R_{q}$, is given by

$\displaystyle
R_{q}=R+2\Delta\psi-\left(1+\frac{1}{q}\right)|\nabla\psi|^{2}.$

By definition, the Ricci flow on $\widetilde{M}$ is defined by

$\displaystyle\partial_{t}\widetilde{g}=-2Ric_{q}.$ (24)

According to Lott [22], the Ricci flow equation $(\ref{WRF-1})$ is equivalent
to the following equations

$\displaystyle\partial_{t}g$ $\displaystyle=$ $\displaystyle-2Ric_{\psi}^{q},$
(25) $\displaystyle\partial_{t}\psi$ $\displaystyle=$
$\displaystyle\Delta\psi-|\nabla\psi|^{2}.$ (26)

Note that, the first equation $(\ref{WRF-2})$ is indeed a modified Ricci flow
equation for the metric $g(t)$ on $M$, and the second one $(\ref{WRF-3})$ is a
forward heat equation for the potential function $\psi(t)$ on $(M,g(t))$. The
systems $(\ref{WRF-2})$ and $(\ref{WRF-3})$ are different from Perelman’s
Ricci flow and the conjugate heat equation introduced in [30], i.e.,

$\displaystyle{\partial g\over\partial t}$ $\displaystyle=$
$\displaystyle-2(Ric+\nabla^{2}f),$ $\displaystyle\frac{\partial f}{\partial
t}$ $\displaystyle=$ $\displaystyle-\Delta f-R,$

and are also different from the following $m$-dimensional Perelman’s Ricci
flow and the conjugate heat equation

$\displaystyle{\partial g\over\partial t}$ $\displaystyle=$
$\displaystyle-2(Ric+\nabla^{2}f-{\nabla f\otimes\nabla f\over m-n}),$
$\displaystyle\frac{\partial f}{\partial t}$ $\displaystyle=$
$\displaystyle-\Delta f+{|\nabla f|^{2}\over m-n}-R.$

Let $\phi$ be a positive solution to the conjugate heat equation on
$(\widetilde{M},\widetilde{g})$

$\displaystyle\partial_{t}\phi=-\Delta_{\widetilde{M}}\phi+R_{q}\phi.$ (27)

Let $\tau\in[0,T]$ be such that

$\displaystyle\partial_{t}\tau=-1.$

Write

$\phi=(4\pi\tau)^{-\frac{n+q}{2}}e^{-\eta}.$

Then

$\displaystyle\partial_{t}\eta$ $\displaystyle=$
$\displaystyle-\Delta_{\widetilde{M}}\eta+|\nabla\eta|^{2}-R_{q}+\frac{n+q}{2\tau},$

Following Perelman [30], the $W$-entropy for the Ricci flow $\widetilde{g}(t)$
on the warped product manifold $\widetilde{M}$ is defined as follows

$\displaystyle W(\widetilde{g},\eta,\tau)$ $\displaystyle=$
$\displaystyle\int_{\widetilde{M}}[\tau(|\widetilde{\nabla}\eta|_{\widetilde{M}}^{2}+R_{q})+\eta-(n+q)]\phi
dvol_{\widetilde{M}},$

where $dvol_{\widetilde{M}}=udvol_{M}dvol_{\mathbb{T}^{q}}$ is volume form on
$(\widetilde{M},\widetilde{g})$.

Applying Perelman’s $W$-entropy formula for the Ricci flow [30] to
$(\widetilde{M},\widetilde{g})$, we have

$\displaystyle\frac{d}{d\tau}W(\widetilde{g},\eta,\tau)=-2\tau\int_{\widetilde{M}}\left|Ric_{q}+\widetilde{Hess}\eta-\frac{\widetilde{g}}{2\tau}\right|_{\widetilde{M}}^{2}\phi
dvol_{\widetilde{M}}.$ (28)

By Theorem 2.1, the Laplace-Beltrami on $(\widetilde{M},\widetilde{g})$ is
given by

$\displaystyle\Delta_{\widetilde{M}}=L+u^{-\frac{2}{q}}\Delta_{\mathbb{T}^{q}},$

where

$\displaystyle L=\Delta-\nabla\psi\cdot\nabla.$

Here $\Delta$ and $\nabla$ are respectively the Laplace-Beltrami operator and
the gradient operator on $(M,g)$. In the case $\phi$ is a function depending
only on the variable of horizontal direction, the conjugate heat equation
$(\ref{PPP4})$ turns out to be the following backward heat equation associated
with the Witten Laplacian on $(M,g(t))$

$\displaystyle\partial_{t}\phi=-L\phi+R_{q}\phi.$ (29)

In this case, $\eta$ is a function depending only on variable in $M$. Thus

$\displaystyle W(\widetilde{g},\eta,\tau)$ $\displaystyle=$
$\displaystyle\int_{M\times\mathbb{T}^{q}}[\tau(|\nabla\eta|^{2}+R_{q})+\eta-(n+q)]\phi
udvol_{M}dvol_{\mathbb{T}^{q}}$ $\displaystyle=$
$\displaystyle\int_{M}\left[\tau(|\nabla\eta|^{2}+R+2\Delta\psi-\left(1+\frac{1}{q}\right)|\nabla\psi|^{2})+\eta-(n+q)\right]\phi
d\mu.$

Here $d\mu=udvol_{M}$, and we assume $vol(\mathbb{T}^{q})=1$. Note that, for
any vector field $v$ on $\widetilde{M}$, by $(\ref{Hess-1})$, $(\ref{Hess-2})$
and $(\ref{Hess-3})$, we have

$\displaystyle\widetilde{Hess}v=Hessv-\frac{1}{q}u^{\frac{2}{q}}\langle\nabla\psi,\nabla
v\rangle\sum^{q}_{\alpha=1}dx^{2}_{\alpha}.$ (30)

Substituting $(\ref{PPP1})$ and $(\ref{PPP3})$ into $(\ref{PPP2})$, we have

$\displaystyle\frac{d}{d\tau}W(\widetilde{g},\eta,\tau)$ $\displaystyle=$
$\displaystyle-2\tau\int_{M}\left|Ric_{\psi}^{q}+Hess\eta-\frac{g}{2\tau}+\frac{1}{q}(\Delta\psi-|\nabla\psi|^{2}-\langle\nabla\psi,\nabla\eta\rangle-{q\over
2\tau})u^{\frac{2}{q}}\sum^{q}_{\alpha=1}dx^{2}_{\alpha}\right|_{\widetilde{M}}^{2}\phi
d\mu$ $\displaystyle=$
$\displaystyle-2\tau\int_{M}\left[\left|Ric_{\psi}^{q}+Hess\eta-\frac{g}{2\tau}\right|^{2}+\frac{1}{q}(\Delta\psi-|\nabla\psi|^{2}-\langle\nabla\psi,\nabla\eta\rangle-{q\over
2\tau})^{2}\right]\phi d\mu.$

Thus we have proved the following $W$-entropy formula for the backward heat
equation associated with the Witten Laplacian on compact manifolds equipped
with Lott’s modified Ricci flow and time dependent potentials.

###### Theorem 5.1

Let $(M,g(t),\psi(t))$ be a compact manifold with a family of Remannian
metrics $g(t)$ and potentials $\psi(t)$ which satisfy

$\displaystyle\partial_{t}g$ $\displaystyle=$
$\displaystyle-2(Ric+Hess\psi-\frac{1}{q}\nabla\psi\otimes\nabla\psi),$
$\displaystyle\partial_{t}\psi$ $\displaystyle=$
$\displaystyle\Delta\psi-|\nabla\psi|^{2}.$

Let $d\mu=e^{-\psi}dvol_{M}$, and $L=\Delta-\nabla\psi\cdot\nabla$. Let $\phi$
be a positive solution to the backward heat equation of the Witten Laplacian
on $M$, i.e.,

$\displaystyle\partial_{t}\phi=-L\phi+R_{q}\phi.$

where $R_{q}=R+2\Delta\psi-(1+\frac{1}{q})|\nabla\psi|^{2}$. Let

$\displaystyle H_{n+q}(\phi,\tau)=-\int_{M}\phi\log\phi
d\mu-\frac{n+q}{2}(1+\log(4\pi\tau)),$

and define the W-entropy $W_{q}(g,\psi,\eta,\tau)$ by

$\displaystyle W_{q}(g,\psi,\eta,\tau)={d\over d\tau}(\tau
H_{n+q}(\phi,\tau)).$

Then

$\displaystyle
W_{q}(g,\psi,\eta,\tau)=\int_{M}[\tau(|\nabla\eta|^{2}+R_{q})+\eta-(n+q)]\phi
d\mu.$

where $\phi=(4\pi\tau)^{-\frac{n+q}{2}}e^{-\eta}$. Moreover

$\displaystyle\frac{d}{d\tau}W_{q}(g,\psi,\eta,\tau)=-2\tau\int_{M}\left[\left|Ric_{\psi}^{q}+Hess\eta-\frac{g}{2\tau}\right|^{2}+\frac{1}{q}\left(\Delta\psi-|\nabla\psi|^{2}-\langle\nabla\psi,\nabla\eta\rangle-{q\over
2\tau}\right)^{2}\right]\phi d\mu.$

In particular, $W_{q}(g,\psi,\eta,\tau)$ is decreasing in the backward time
$\tau$, and the monotonicity is strict unless that

$\displaystyle Ric_{\psi}^{q}+Hess\eta$ $\displaystyle=$
$\displaystyle\frac{g}{2\tau},$ $\displaystyle\Delta\psi-|\nabla\psi|^{2}$
$\displaystyle=$ $\displaystyle\langle\nabla\psi,\nabla\eta\rangle-{q\over
2\tau}.$

As an application of the $W$-entropy formula in Theorem 1.3, we have the
following result, which is the restatement of Perelman’s monotonicity theorem
for the $\mu$-invariant for Ricci flow [30] on warped product manifolds. See
also [7, 10, 15, 25].

###### Theorem 5.2

Let $(M,g(t),\psi(t))$ be a compact manifold with a family of Remannian
metrics $g(t)$ and potentials $\psi(t)$ which satisfy

$\displaystyle\partial_{t}g$ $\displaystyle=$
$\displaystyle-2(Ric+Hess\psi-\frac{1}{q}\nabla\psi\otimes\nabla\psi),$
$\displaystyle\partial_{t}\psi$ $\displaystyle=$
$\displaystyle\Delta\psi-|\nabla\psi|^{2}.$

Then there exists a positive and smooth function $u=e^{-{\eta\over 2}}$ such
that $\eta$ achieves the optimal logarithmic Sobolev constant $\mu(\tau)$
defined by

$\displaystyle\mu(\tau):=\inf\limits\left\\{W_{q}(g,\psi,\eta,\tau):\
\int_{M}{e^{-\eta}\over(4\pi\tau)^{n+q\over 2}}d\mu=1\right\\}.$

where

$\displaystyle
W_{q}(g,\psi,\eta,\tau)=\int_{M}[\tau(|\nabla\eta|^{2}+R_{q})+\eta-(n+q)]\phi
d\mu,$

Indeed, $u=e^{-{\eta\over 2}}$ is a solution to the nonliniear PDE

$\displaystyle-4\tau Lu+\tau R_{q}u-2u\log u-(n+q)u=\mu(\tau)u.$

Moreover, $\mu(\tau)$ is decreasing in $\tau$ on $[0,T]$.

Acknowledgement. The second author would like to thank Bing-Long Chen for
useful discussion in January 2006.

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Songzi Li

School of Mathematical Science, Fudan University, 220, Handan Road, Shanghai,
200432, China

Xiang-Dong Li

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55,
Zhongguancun East Road, Beijing, 100190, China, E-mail<EMAIL_ADDRESS>