art%3A10.1007%2Fs00013-011-0263-0

Arch. Math. 96 (2011), 473–481c! 2011 Springer Basel AG0003-889X/11/050473-9published online May 13, 2011DOI 10.1007/s00013-011-0263-0 Archiv der Mathematik

A new entropy formula for the linear heat equation

Lin-Feng Wang

Abstract. We give a monotonicity entropy formula for the linear heatequation on complete manifolds with Ricci curvature bounded frombelow. As its applications, we get a differential Harnack inequality and alower bound estimate about the heat kernel.

Mathematics Subject Classification (2000). Primary 53C65;Secondary 58J35.

Keywords. Heat kernel, Entropy, Differential Harnack inequality.

1. Introduction. Let M be an n-dimensional complete manifold with metric gfixed or evolving along the Ricci flow. Define

W (f, !) =!

M

(! |!f |2 + f " n)e!f

(4"!)n2

dV , (1.1)

where ! > 0 and (f, !) satisfies!

M

e!f

(4"!)n2

dV = 1. (1.2)

Let

µ(!) = inf{W (f, !) :!

M

e!f

(4"!)n2

dV = 1}. (1.3)

In [4], Perelman ruled out nontrivial shrinking breathers on closed mani-folds by using the monotonicity property of µ(!) along the Ricci flow. From

This work was completed with the support of the NSF of China (10871070, 10971066).

474 L.-F. Wang Arch. Math.

then on, the entropy formula turned out to be of the fundamental importancein the study of the Ricci flow. Later, Ni considered the linear heat equation

"#

#t" #

#u(x, t) = 0 (1.4)

on manifolds with fixed metric in [2,3]. The main result in [2] states thatW (f, !) is monotone non-increasing along the heat equation (1.4) whenthe Ricci curvature is nonnegative. Based on this monotone result, Ni gota differential Harnack inequality for the heat kernel, he also proved thatlim!"0 µ(!) = 0 on closed manifolds with nonnegative Ricci curvature.

For K $ 0, we define

WK(f, !) =!

M

$e2K! " 1

2K|!f |2 + f +

n

2

"log

2K!

e2K! " 1" e2K! " 1

2K

#" n

%

% e!f

(4"!)n2

dV, (1.5)

where ! > 0 and (f, !) satisfies (1.2). In Section 2, we prove a monotone for-mula for WK(f, !) along the heat equation (1.4) on closed manifolds with Riccicurvature bounded from below by "K. As an application, we get a differentialHarnack inequality of the heat kernel, which can be regarded as a generalizedLaplacian comparison theorem in space-time.

Define

µK(!) = inf{WK(f, !) :!

M

e!f

(4"!)n2

dV = 1}. (1.6)

In Section 3, we show that on manifolds with Ric $ "K, µK(!) is a monotonicdecreasing function of ! . Moreover, lim!"0 µK(!) = 0. In Section 4, we usethe monotonicity formula to estimate the heat kernel from below.

2. Monotonicity formula. Let M be a closed manifold, we assume that u =e!f

(4"!)n2

is a positive solution of the equation (1.4) with&

M u dV = 1. Then wecan prove the following result.

Theorem 2.1. WK(f, !) satisfies

dWK

dt="

!

M

e2K! " 1K

$|!i!jf" Ke2K!

e2K! " 1gij |2+Rijfifj + K|!f |2

%u dV ,

(2.1)

whered!dt = 1. In particular, WK(f, !) is monotonic non-increasing along the

heat equation (1.4) when Ric $ "K.

Proof. Let u = e!f and w = 2#f " |!f |2. Define

UK =e2K! " 1

2Kw + f +

n

2

$log

K

2"(e2K! " 1)" (e2K! " 1)

%" n.

Vol. 96 (2011) A new entropy formula for the linear heat equation 475

From the proof of Lemma 2.1 in [2], we have"#

#t" #

#w = "2(|!i!j f |2 + Rij fifj) " 2 < !w,!f > .

Hence,"#

#t" #

#UK = e2K!w +

e2K! " 12K

#w

#t

+#f

#t" n

2

"2Ke2K!

e2K! " 1+ 2Ke2K!

#" e2K! " 1

2K#w " #f

& e2K! " 12K

("2|!i!j f |2 " 2Rij fifj) " 2 < !w,!f >)

+e2K!wi " |!f |2 " n

2

"2Ke2K!

e2K! " 1+ 2Ke2K!

#,

which means that"#

#t" #

#(UKu)

& e2K! " 12K

("2|!i!j f |2 " 2Rij fifj) " 2 < !w,!f >)u

+e2K!wu " |!f |2u " n

2

"2Ke2K!

e2K! " 1+ 2Ke2K!

#u " 2

<e2K! " 1

2K!w + !f ,!u >

= "e2K! " 1K

|!i!jf " Ke2K!

e2K! " 1gij |2u " e2K! " 1

K[Rijfifj + K|!f |2]u.

Integrating above inequality on M yields (2.1). !

The following differential Harnack inequality can be seen as an applicationof Theorem 2.1.

Corollary 2.2. Let M be a closed manifold with Ric $ "K. We use H(x, y, t)to denote the heat kernel. Thene2Kt " 1

2K(2#f " |!f |2) + f +

n

2

$log

2Kt

e2Kt " 1" (e2Kt " 1)

%" n & 0, (2.2)

where f = " log (4"t)n2 H.

Proof. We first prove that if u is the fundamental solution of (1.4), then'

(!

M

hUKu dV

)

*

++++++!=0

= 0 (2.3)

holds for any positive smooth function h on M . In fact, let e2K! !12K = $, then

UK(f, !) = U(f, $) +n

2log

log (1 + 2K$)2K$

" nK$,


where

U = !(2#f " |!f |2) + f " n.

It was proved in [2] that if u is the fundamental solution of (1.4), then'

(!

M

hUu dV

)

*

++++++!=0

= 0

holds for any positive smooth function h on M , hence we get (2.3).For t0 > 0 and any positive smooth function % on M , we solve the backward

heat equation"#

#t+ #

#%t = 0

with initial value %t0 = %, then

ddt

!

M

%tUKH dx =!

M

#

#t%tUKH + %t

#

#t(UKH) dx

=!

M

"#

#t+ #

#%tUKH + %t

"#

#t" #

#(UKH) dx

& 0.

Due to (2.3), we know that for t0 > 0 and any positive smooth function %,!

M

%UKH dx & 0,

which means UKH & 0, or UK & 0. !

Remark 2.3. On a manifold with nonnegative Ricci curvature, Li-Yau’s gradi-ent estimate([1])

ut

u" |!u|2

u2+

n

2t$ 0

is equivalent to the differential inequality 2t#f & n, where f = " log (4"!)n2 u.

Ni Lei got his differential inequality about the heat kernel on a closed mani-fold with nonnegative Ricci curvature. (2.2) can be regarded as a generalizedLaplacian comparison theorem in space-time. In fact, let L = 1!e!2Kt

2K f , (2.2)can be rewritten as

#L + Lt & e!2Kt

$n " n

2log

2Kt

e2Kt " 1" n

2(e2Kt " 1) " n(e2Kt " 1)

4Kt

%.

If K ' 0, we recover Ni’s generalized Laplacian comparison theorem in space-time, see Theorem 1.2 in [2].


3. Property of µK(! ). The monotonicity property of µ(!) plays an importantrole when Perelman tries to rule out nontrivial shrinking breathers on closedmanifolds [4]. On manifolds with nonnegative Ricci curvature, Ni proved thatµ(!) is a monotone decreasing function of ! , he also got lim!"0 µ(!) = 0 in[2]. In this section, we first give a generalization of Ni’s result about µ(!), ofindependent interest.

Lemma 3.1. Let M be a closed manifold, then

lim!"0

µ(!) = 0. (3.1)

Proof. We only need to prove that lim sup!"0 µ(!) & 0. The rest part is similarto the proof of Corollary 1.3 in [2]. By Theorem 1.1 in [2],

#W

#t= "

!

M

2!(|!i!jf " 12!

gij |2 + Rijfifj)u dV .

If we assume that the Ricci curvature is bounded from below by "K0 withsome constant K0 $ 0, then

#W

#t& 2K0!

!

M

|!f |2u dV = 8K0!

!

M

|!v|2 dV

and

W =!

M

(4! |!v|2 " v2 log v2) dV " n

2log (4"!) " n,

where

u =e!f

(4"!)n2

= v2

is the positive heat kernel. By Jensen’s and Sobolev’s inequalities, we cancompute as follows,

"!

M

v2 log v2 dV = "n " 22

!

M

log v4

n!2 v2 dV

$ "n " 22

log

'

(!

M

v4

n!2+2 dV

)

*="n " 22

log

'

(!

M

v2n

n!2 dV

)

*

$ "n " 22

log

,

-.C

'

(!

M

(|v|2 + |!v|2) dV

)

*

nn!2/

01

= "n " 22

log C " n

2log

'

(1 +!

M

|!v|2 dV

)

*, (3.2)


where C is the Sobolev constant of M . For convenience, we denote&

M |!v|2dV = A, then for every x > 2K0,

#W

#t& xW + (8K0 " 4x)!A +

nx

2log (4"!(1 + A)) +

n " 22

x log C + nx.

It is easy to testify that

(8K0"4x)!A+nx

2log (1+A)& nx

2

"log

nx

8(x " 2K0)!" 1#

+ 4!(x " 2K0).

Hence#W

#t& xW + 8(x " 2K0)! + G(x),

where

G(x) =nx

2

"1 + log

n"x

2(x " 2K0)

#+

n " 22

x log C.

So we get

#

#!

$e!x! (W +

8!(x " 2K0)x

+8(x " 2K0)

x2+

G(x)x

)%

& 0.

Note that u is the positive heat kernel, so W (f, !)|!=0 = 0 and we have

W (f, !) &"

8(x " 2K0)x2

+G(x)

x

#(ex! " 1) " 8!(x " 2K0)

x,

which means that

µ(!) &"

8(x " 2K0)x2

+G(x)

x

#(ex! " 1) " 8!(x " 2K0)

x.

We conclude that

lim sup!"0

µ(!) & 0.

!

Now we can prove the following theorem.

Theorem 3.2. Let M be a closed manifold with Ric $ "K. Then µK(!) is amonotone decreasing function of ! . Moreover

lim!"0

µK(!) = 0. (3.3)

Proof. The monotone property of µK(!) is a natural corollary of Theorem 2.1.It is easy to see that µK(!) & 0 by (2.2). Due to Lemma 3.1, in order to prove(3.3), we only need to show that

µ(!) & µK(!) " n

2

$log

2K!

e2K! " 1" (e2K! " 1)

%. (3.4)


By [5], we can choose f to be a smooth minimizer of WK(f, !) for some ! > 0,then

W (f, !) & WK(f, !) " n

2

$log

2K!

e2K! " 1" (e2K! " 1)

%

= µK(!) " n

2

$log

2K!

e2K! " 1" (e2K! " 1)

%.

Hence (3.4) follows. !

4. Heat kernel estimate. As an application of the differential Harnack inequal-ity, we prove a lower bound estimate of the heat kernel.

Theorem 4.1. Let M be a closed manifold with Ric $ "K. We use H(x, y, t)to denote the heat kernel. Then

!

M

log

2H

"2e"(e2Kt " 1)

K

#n23

H dV $ 0 (4.1)

and

H(x, x, 2t) $"

2e"(e2Kt " 1)K

#! n2

. (4.2)

Proof. Let

f = " log (H(x, y, t)(4"t)n2 )

and

g =1 " e!2Kt

2K

$f " n

2

"1 + log

e2Kt " 12Kt

#%.

By using (2.2) and the fact that

ft = #f " |!f |2 " n

2t,

we can compute as follows,

#g + gt =1 " e!2Kt

2K(2#f " |!f |2)

+e!2Ktf " n

2e!2Kt

"1 + log

e2Kt " 12Kt

#& 0,

soddt

!

M

Hg dV =!

M

(#Hg + gtH) dV

=!

M

(#g + gt)H dV & 0.

By [6], for some fixed point y ( M , if x /( Cut(y), then

limt"0

(tf(x, t)) =d2(x, y)

4,


where d(x, y) denotes the geodesic distance between x and y, which meansthat

limt"0

!

M

Hg dV = 0,

hence !

M

gH dV & 0,)t > 0

and (4.1) follows. Noting that log x is concave, by using Jensen’s inequality,we get

!

M

H2 dV $"

2e"(e2Kt " 1)K

#! n2

,

the semigroup property of the heat kernel H(x, y, t) tells us that (4.2) is right.!

Remark 4.2. If M is a closed manifold with nonnegative Ricci curvature,H(x, y, t) is the heat kernel, then (4.1) and (4.2) becomes

!

M

log (H(4"te)n2 )H dV $ 0 (4.3)

and

H(x, x, 2t) $ (4"te)! n2 .

The heat kernel on Rn is

H(x, y, t) =1

(4"t)n2

e! d2(x,y)4t .

A little computation shows that (4.3) becomes equality when M = Rn. Weonly testify this fact when n = 1, which is the following equality

#!

!#

1*4"t

x2

4te! x2

4t dx =*"

2.

Acknowledgements. The author would like to thank the referee for valuablecomments. The author is also grateful to his advisor professor Chunli Shen forhis constant encouragement.

References

[1] P. Li and S. T. Yau, On the parabolic kernel of the Schrodinger operator, Acta

Math. 156 (1986), 153–201.

[2] L. Ni, The entropy formula for linear heat equation, J. Geom. Anal. 14 (2004),

87–100.

[3] L. Ni, Addenda to “The entropy formula for linear heat equation”, J. Geom.

Anal. 14 (2004), 329–335.


[4] G. Perelman, The entropy formula for the Ricci flow and its geometric appli-

cations, arXiv:math.DG/0211159.

[5] S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrodinger

operators, J. Funct. Anal. 42 (1981), 110–120.

[6] D. Stroock and J. Turetsky, Short time behavior of logarithmic derivatives

of the heat kernel, Asian J. Math. 1 (1997), 17–33.

Lin-Feng WangSchool of Science,Nantong University,Nantong 226007, Jiangsu,People’s Republic of Chinae-mail: [email protected]; [email protected]

Received: 4 September 2010

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