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The entropy formula for linear heat equation

We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman's recent results on volume non-collapsing.

In a recent paper of Perelman [P1], an entropy formula for Ricci flow was derived. The formula turns out being of fundamental importance in the study of Ricci flow (cf. [P1,Sections 3,4,10]) as well as the Kähler-Ricci flow [P2]. The derivation of the entropy formula in [P1,Section 9] resembles the gradient estimate for the linear heat equation proved by Li-Yau in another fundamental paper [L-Y] on the linear parabolic equation. This suggests that there may exist a similar entropy formula for the linear heat equation. The purpose of this short note is to show such entropy formula and derive some applications of the new entropy formula.
Let M be a complete Riemannian manifold. We study the heat equation Following [P1], we define In particular, if M has nonnegative Ricci curvature, W(f, τ ) is monotone decreasing along the heat equation.
Notice that in the case that M is Ricci flat, the result above is indeed a special case of Perelman's result. We just show that it in fact also holds for all metrics with nonnegative Ricci curvature.
The result can be derived out of a point-wise differential inequality. The proof of Theorem 0.1 and the argument of [P1] gives the following differential inequality for the fundamental solution to the heat equation.  Notice that Li-Yau's gradient estimate u t u − |∇u| 2 u 2 + n 2t ≥ 0 is equivalent to (0.6) t(2∆f ) − n ≤ 0.
The inequality (0.6) can be viewed as a generalized Laplacian comparison theorem. In deed, the Laplacian comparison theorem on M is a consequence of (0.6) by applying the inequality to the heat kernel and letting t → 0. This suggestions that one can view L(x, t) = 4tf as the square of a time-dependent 'distance function'. Then (0.6), which says that ∆L ≤ 2n, simply generalizes the standard Laplacian comparison ∆r 2 ≤ 2n on any Ricci non-negative manifold to such generalized 'distance function'. From this point of view, one can think (0.5) as a Laplacian comparison theorem in the space-time since it says ∆L +L t ≤ 2n. The similar inequality [P1, (7.15)] was one of the important new discoveries of Perelman. Applying similar consideration, one can also view the entropy estimate in [P1] as a generalization of the space-time Laplacian comparison theorem. This is also related to the reduced volume monotonicity of [P1]. It was pointed to us later by Professor S. T. Yau that Hamilton and him also noticed (0.5) for the Ricci flow a few years ago. It seems that (0.5) and (0.6) do not imply each other. It is interesting to find out if there is any deeper connection between them. For closed manifolds, following [P1], one can define A direct consequence of Theorem 0.1 is the following Corollary 0.1. On manifolds with nonnegative Ricci curvature, µ(τ ) is a monotone decreasing function of τ . Moreover µ(τ ) ≤ 0 with lim τ →0 µ(τ ) = 0.
Whenever it makes sense (for example, when M is simply-connected, negatively curved with lower bound on its curvature), as in [P 1] one can also define ν = inf τ µ(τ ). It can be thought as some sort of isoperimetric constant. When M = R n , ν = 0. Thanks to the gradient estimates of Li-Yau [L-Y], the above results still hold on complete noncompact manifolds with nonnegative Ricci curvature. As an application of the entropy formula obtained in Theorem 0.1 we prove the following result.
Theorem 0.3. Let M be a complete Riemannian manifold with nonnegative Ricci curvature. Then µ(τ ) ≥ 0 for some τ > 0 if and only if M is isometric to R n .
In [G] (see also [S, W]), a sharp logarithmic Sobolev inequality (in different disguises in [S, W]) was proved on R n . When M = R n , the inequality is equivalent to Since (0.8) is equivalent to µ( 1 2 ) ≥ 0, a simple corollary of Theorem 0.3 is the following result on the relation between the logarithmic Sobolev inequality and the geometry of the manifolds, which is originally due to Bakry, Concordet and Ledoux [B-C-L]. It can be shown that the (0.8) holds on any manifold with sharp isoperimetric inequality, or equivalently the sharp L 1 -Sobolev inequality. Under the request of some readers of the preliminary version, we include a proof of this fact (cf. Proposition 3.1) using the spherical symmetrization. The proof was communicated to the author by Perelman last year in November. The proof only uses the spherical symmetrization to compare with the Euclidean case. One can find the simple elegant proof of (0.8) by Beckner and Pearson in [B-P], which makes use of the fact that the product of Euclidean spaces is still Euclidean together with the sharp L 2 -Sobolev inequality.
It turns out that W(f, τ ) being finite, where u = e −f (4πt) n 2 is the heat kernel, also has strong geometric and topological consequences. For example, in the case M has nonnegative Ricci curvature, it implies that M has finite fundamental group. In fact we can show that M is of maximum volume growth if and only if the entropy W(f, t) is uniformly bounded for all t > 0, where u = e −f (4πt) n 2 is the heat kernel. The analogy of above was discovered originally in [P1] for the ancient solution to Ricci flow with bounded nonnegative curvature operator, where he claimed that an ancient solution to the Ricci flow with nonnegative curvature operator is κ-non-collapsed if and only if the entropy is uniformly bounded for any fundamental solution to the conjugate heat equation.
Without assuming the nonnegativity of the Ricci curvature, the bound on µ(τ ) also implies the uniform lower bound on the volume of balls of certain scales. Namely, it implies the volume noncollapsing, as in the κ-noncollapsing theorem of Perelman [Theorem 4.1,P1], therefore an uniform upper bound of the diameter, if the manifold has finite volume. In some sense, µ(τ ) reflects the isoperimetric property of M for the scale parametrized by τ .
Acknowledgement. The author would like to thank Professor Ben Chow for encouragement to study Perelman's recent papers and discussions, Professor Peter Li for helpful suggestions. Special thanks goes to Professor Perelman since most results in this paper are the simpler versions of the corresponding results in [P1] without Ricci flow. §1 Proof of Theorem 0.1 and 0.2.
We start with the following two lemmas.
Lemma 1.1. Let M be a complete Riemannian manifold. Let u be a positive solution to (0.1). Then Proof. Direct calculation shows that Here we have used w = 2f t + |∇f | 2 and ∂ ∂t − ∆ f = −|∇f | 2 . Lemma 1.2. Let M and u be as in Lemma 1.
Proof. One can proceed directly. Here we use Lemma 1.1 to simplify the calculation a little. Letf = − log u. Then we have that Here we have used ∇f = ∇f .
Remarks. 1. Lemma 1.1 has its corresponding version for the Ricci flow. Namely, if g ij satisfies the back-ward Ricci flow equation Here u = e −f . One can easily see that (1.3) implies the formula (1.2) of [P1]. This also gives another derivation of the first monotonicity formula in [P1]. 2. The above approach of the proof to Lemma 1.2 was motivated by the statistical analogy in [P1,Section 6]. One can also use the similar approach to derive Proposition 9.1 of [P1] from (1.3) above. This would simplify the calculation a little and reflect the relation between the energy and the entropy quantity.
Proof of Theorem 0.1. The proof of Theorem 0.1 follows from the simple observation u∇f = −∇u, therefore and integration by parts.
Proof of Theorem 0.2. We can apply Perelman's argument in the proof of Corollary 9.3 of [P1]. For any t 0 > 0, let h be any positive function. We solve the backward heat equation starting from t 0 with initial data h. Then we have that Using the fact that ( M hW u dv)| t=0 = 0, when u is the fundamental solution, we have that for any t 0 > 0 and any positive function h. This implies that W u ≤ 0. Therefore W ≤ 0. §2 Extensions and the value of µ(0).
The first extension is to complete noncompact manifolds. From the proof, it is easy to see that Theorem 0.1 and 0.2 hold as long as the integration by parts can be justified. We focus on the case M has nonnegative Ricci curvature. Since we have the gradient estimate of Li-Yau for the positive solutions one indeed can make the integration by part rigorous, keeping in mind that u is assumed integrable in our consideration of the entropy. One of the reference where one can find the estimates on derivatives of u is [C-N, Section 3].
Another extension of Theorem 0.1 and Theorem 0.2 is for manifolds with boundary. In this case, it is not hard to show that the theorem holds on manifolds with convex boundary. In fact, in this case Here h ij denotes the second fundamental form of ∂M . Therefore, Theorem 0.1 and Theorem 0.2 hold for positive solution u with the Neumann boundary condition ∂u ∂ν = 0. Corollary 2.1. Let M be a compact manifold with boundary. Let u be a positive solution to (0.1) with the Neumann boundary condition. Let f and τ be as in Theorem 0.1. Then Here II(·, ·) is the second fundamental form of ∂M and (∇f ) T is the tangential projection of ∇f on ∂M . In particular, in the case M has nonnegative Ricci and ∂M is convex, W is monotone decreasing. Moreover, if u is the fundamental solution, for t > 0.
Since we know that µ(τ ) is monotone, it is nice to know the value of µ(τ ) as τ → 0. We can adapt the argument of [P1] to prove that lim τ →0 µ(τ ) = 0. Since the argument in [P1] of this part is sketchy we include a detailed proof for the sake of the reader. But the original idea is certainly from [P1].
Proposition 2.1. Let M be a closed manifold.
Proof. It is easy to see that µ(τ ) ≤ 0 by Theorem 0.2. Assume that there exists τ k → 0 such that µ(τ k ) ≤ c < 0 for all k. We show that this will contradict the logarithmic Sobolev inequality. We are going to blow up the metric by 1 2 τ −1 . First we can decompose M into open subsets U 1 , U 2 , · · · , U N such that each U j is contained inside some normal coordinates and each U j also contains B(o j , δ), a ball of radius δ, for some small δ > 0. Now let g τ = 1 2 τ −1 g ij and g k = g τ k . It is clear that (U j , g k , o j ) converges to (R n , g 0 , 0) in C ∞ norm. We will also identify the compact subset of R n with the compact subset of U j .
It is easy to see that n 2 dv τ = 1, where | · | τ is the norm with respect to g τ = 1 2 τ −1 g and dv τ is the corresponding volume form. It is also convenient to write in more standard form: restricted to M ψ 2 dv τ = 1. Let ϕ k be the minimizer realizing µ(τ k ). Then we have that Here ∆ k denote the Laplacian of g τ k and dv k = dv τ k . Due to the monotonicity, we can also assume that µ(τ k ) ≥ −A for some A > 0 independent of k. Now we write F k (ψ) = 2|∇ψ| 2 τ k − (log ψ 2 )ψ 2 − ( n 2 log(2π) + n)ψ 2 . It is a easy matter to check that By the assumption that µ(τ k ) ≤ c < 0 we know that M F (ϕ k ) dv k ≤ c < 0.
By passing to subsequence we can assume that It is easy to see that U 1 ϕ 2 k dv k ≤ 1. Combining the above with (2.5) and the fact that g k converges to g 0 on every fixed compact subset of R n , the elliptic PDE theory implies that there exists a subsequence of ϕ k (still denote by ϕ k ) such that it converges uniformly on every compact subset of R n . If the limit function ϕ ∞ exists and R n ϕ 2 ∞ dv 0 > 0 we claim that we will get contradiction to the logarithmic Sobolev inequality (0.8). In fact in this case we just denote ǫ 2 = R n ϕ 2 ∞ . Clearly 0 < ǫ ≤ 1 by the assumption. Since This is a contradiction to the sharp logarithmic Sobolev inequality (0.8). On the other hand, if ǫ = 0. which would imply ϕ ∞ = 0. This contradicts (2.8). We therefore complete the proof of the proposition. §3 Bounded entropy and volume growth.
The main purpose of the section is to prove Theorem 0.3 and show that for the manifold with nonnegative Ricci curvature the finiteness of the entropy for the heat kernel is equivalent to the manifold has maximum volume growth. We first include a short discussion on the logarithmic Sobolev inequality. We say M has logarithmic Sobolev inequality if n 2 dv = 1. This is equivalent to the finiteness of µ( 1 2 ). It is an easy matter to see that the regular Sobolev inequality implies (3.1). In particular, it holds on minimal submanifolds in R n , which is a special case of the general result in [E]. Since one has the L 2 -Sobolev inequality on a closed manifold, any closed manifold satisfies (3.1). In this case the dependence of the constant C 1 can be explicitly traced, applying Lemma 2 of [L1]. This would in turn gives the explicit dependence of the κ constant on the geometry of the initial metric in the κ noncollapsing theorem of [P1,Section 4].
The inequality (3.1) is also equivalent to the ultracontractivity as pointed out in [D], which then follows from conditions such as the lower bound on the Ricci curvature (i.e. Ric ≥ −K for some K ≥ 0) and inf V x (1) ≥ δ > 0. Therefore, µ( 1 2 ) is finite for a large class of manifolds.
It is interesting to find out on which manifolds the logarithmic Sobolev inequality holds with C 1 = n. Namely (0.8) holds. It was pointed out in [P1] that the sharp isoperemetric inequality also implies the sharp logarithmic Sobolev inequality. The following was the proof suggested by the communication with Perelman. The result was somewhat conjectured in [R].
It suffices to prove the result for compact supported nonnegative function φ. Let M ′ = {x|φ(x) > 0}. LetB(R) be a ball of radius R in R n such that V ol(B(R)) = V ol(M ′ ). We define that F (t) = V ol({x ∈ M ′ |φ(x) ≥ t}). We also denote M t = {x ∈ M ′ |φ(x) ≥ t}. Γ t = ∂M t . Let g(|y|) be a function onB (R) such that V ol({y|g(y) ≥ t}) = F (t) and g(R) = 0. We can defineM t andΓ t similarly. Clearly V ol(M t ) = V ol(M t ) and A(Γ t ) ≥ A(Γ t ) by the isoperimetric inequality. The simply integration by parts shows that for any Lipschitz function λ(t) with λ(0) = 0. This implies that On the other hand the isoperimetric inequality implies ) |∇g| 2 dv.
In fact, the co-area formula shows that and Combining with the fact dĀ.
Using the fact that |∇g| is constant onΓ t , by Hölder inequality, we have that Since (3.4) implies (3.5) we completes the proof.
We should point out that in [B], Beckner provides a proof of (0.8) from the isoperimetric inequality using the product structure of the Euclidean spaces. The above argument just reduces the (0.8) for any manifolds, with the sharp isoperimetric inequality, to the Euclidean space (with the same dimension) case. It does not prove the Euclidean case itself. Now we prove Theorem 0.3.
Combining with the assumption that Ricci is nonnegative this implies that M is isometric to R n . In fact from (3.7) one can easily obtain that where A(r) and V (r) denotes the area of ∂B x (r) and the volume of B x (r), which then implies that V Since we only uses the behavior of the heat kernel near t = 0, our proof is dual to theirs in some sense. In [Le], the author proved that the sharp Sobolev inequalities on a Ricci nonnegative manifold also implies the manifold is isometric to R n . The case of L 1 -Sobolev, which is equivalent to the isoperimetric inequality in Proposition 3.1, is relatively simple. The other cases are more involved. Please see [Le] for details. It was also asked in [Le] if the sharp Nash inequality implies the same conclusion or not. That still remains open.
Proposition 3.2. Let M be a complete Riemannian manifold with nonnegative Ricci curvature. Assume that M has maximum volume growth, namely V o (r) r n ≥ θ for some θ > 0. Then there exists A = A(θ, n) > 0 such that n 2 being the heat kernel. On the other hand, (3.8) implies that M has maximum volume growth. Namely V o (r) r n ≥ θ holds for some θ = θ(n, A).
On the other hand, by Li-Yau's heat kernel estimate Here we have used the fact M v 2 dv = 1.
To prove the second half of the claim we need to use the lower bound estimate of Li-Yau as well as the gradient estimate for the heat kernel. We first estimate the first term in (3.9) using inequality (0.6), the Li-Yau's gradient estimate. (3.10) The second term can be estimated as (3.11) Here C i are positive constants only depending on n. We also have used Theorem 3.1 of [N] to estimate the last term of the second line above. Putting the assumption W ≥ −A and (3.9)-(3.11) together we have the lower bound (3.1) for the volume.
The similar result as above was claimed in [P1,Section 11] for the Ricci flow ancient solutions. The proof here is easier than the nonlinear case considered in [P1]. In fact, Proposition 3.2 here can be used in the proof of Theorem 10.1 of [P1]. §4 Manifolds with bounded µ(τ ).
The following result gives the geometric implication of the non-sharp logarithmic Sobolev inequality (3.3), or bounded µ(τ ). The result can be thought as Riemannian version of the κ non-collapsing result of [P1]. Notice that we do not even require M has nonnegative Ricci curvature. The results in this section are in the line of Perelman's work on Kähler-Ricci flow [P2]. However, the arguments in the nonlinear case are technically more involved than the case treated here, especially on the diameter bound. Proof. The first observation is that for compact supported nonnegative function h. If we have that for η = 1 3 n we will have the estimate (4.1). The reasoning is exactly as in [P1], by n 2 ζ 2 (r x (y)/R), where ζ be a nonnegative cut-off function such that ζ(t) = 1 for all t ≤ 1 2 , and ζ(t) = 0 for t ≥ 1. B is so chosen such that M h 2 dv = 1. Under the assumption (4.3) we have that Therefore, estimation on the right hand side of (4.2) gives which implies (4.1) for some κ. Now argue by contradiction that (4.1) must holds. If not, we know that (4.3) can not be true. Namely We focus on the smaller ball B x ( R 2 ). By the above argument we would conclude that by the assumption (4.1) does not hold. Therefore, iterating the argument we have that for all natural numbers k. This leads to V x (r) ≤ Cr n log 2 3 for small r, which is a contradiction.
The following result on the diameter of a manifold with bounded µ(τ ) is an easy consequence of Theorem 4.1. In fact, D ≤ 2([ V 0 κ ] + 1). In particular, it implies that M is compact if it is complete. Concluding remarks. 1) It would be interesting to find out if there is an interpolation between the entropy formula of Perelman and (0.4). Namely to find a family monotonicity formulae connecting both. For the differential Harnack, or Li-Yau-Hamilton inequality, there is such interpolation in dimension two as shown by Chow [Ch]. The straightforward formulation seems not to work. (One could have some differential inequalities connecting both cases. But the differential inequalities does not give monotonicity formulae unless on two end points.) 2) It seems that the entropy formula in [P1] is essentially different from the known one of Hamilton [H1] for the Ricci flow on Riemann surfaces since it can be used to derive the uniform scalar curvature bound and diameter bounds without appealing the Harnack inequality (in [P2] Perelman proved these results for Kähler-Ricci flow with c 1 (M ) > 0), unlike the approach in [H1], which used the Harnack inequality for the Ricci flow essentially (in [C-C-Z], using the similar method of Hamilton on Riemann surfaces, the authors proved the scalar curvature and diameter bound for the case when the manifolds has positive bisectional curvature, which is a special, relatively easier, case of what treated in [P2]). Is there any connection between Perelman's entropy formula and Hamilton's entropy formula at all?
3) Whether Theorem 0.3 is still true in n = 3 by assuming instead the scalar curvature R(x) of M is nonnegative? 4) In [C-N], the authors proved the matrix Li-Yau-Hamilton inequality on Kähler manifolds with nonnegative bisectional curvature following an earlier work of Hamilton [H2], which can be viewed as a generalized complex Hessian comparison theorem. The natural question is: does (0.2) have a matrix version? The same question applies to Perelman's entropy estimate Corollary 9.3.