Sharp Hamilton's Laplacian estimate for the heat kernel on complete manifolds

In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an estimate on Laplacian form of the heat kernel on complete manifolds with nonnegative Ricci curvature that is sharp in the order of time parameter for the heat kernel on the Euclidean space.


Introduction
In [4], R. Hamilton established the following gradient and Laplacian estimates for a bounded positive solution to the heat equation on closed manifolds with Ricci curvature bounded below. Theorem 1.1 (R. Hamilton [4]). Let (M, g) be an n-dimensional closed Riemannian manifold with Ricci curvature satisfying Ric ≥ −K for some constant K ≥ 0.  If we further assume T ≤ 1, then for 0 ≤ t ≤ T , For estimate (1.2), when K = 0, by choosing function ϕ = t in Hamilton's proof of Lemma 4.1 in [4], we easily confirm that the condition "T ≤ 1" can be removed. Hamilton's estimate (1.1) shows that one can compare two different points at the same time, however the well-known Li-Yau's gradient estimate [7] only allows comparisons between different points at the different times. In [6], B. Kotschwar generalized the gradient estimate (1.1) to the case of complete, noncompact Riemannian manifolds with Ricci curvature bounded below. Using this generalization, Kotschwar gave an estimate on the gradient of the heat kernel for complete manifolds with nonnegative Ricci curvature. Moreover this estimate is sharp in the order of t for the heat kernel on R n .
Kotschwar's result can be used to prove the monotonicity of Ni's entropy functional (stationary metric for Perelman's W-functional in [10]) in [8] for the fundamental solution to the heat equation on complete, noncompact manifolds. We would like to point out that, in the course of justifying the monotonicity Ni's entropy functional on complete noncompact manifolds, one may need a noncompact version of Hamilton's Laplacian estimate (1.2). However, as far as we know, perhaps no people generalized the estimate (1.2) to the complete noncompact case. In [2], the authors only briefly sketched a proof of the estimate (1.2) for the noncompact case but didn't give any detail. In this paper, we will provide a full detailed proof that Hamilton's Laplacian estimate (1.2) also holds for complete, noncompact manifolds with nonnegative Ricci curvature. Precisely, we show that Theorem 1.2. Let (M, g) be an n-dimensional complete noncompact Riemannian manifold with nonnegative Ricci curvature. Suppose u is a smooth positive solution to the heat equation for all x ∈ M and 0 ≤ t ≤ T .
The proof of Theorem 1.2 is similar to the arguments of Kotschwar [6], which can be divided into two steps. In the first step, we obtain some Bernstein-type estimate of ∆u, similar to the upper estimate of |∇u| derived by Kotschwar [6] on complete noncompact manifolds. In the second step, using upper estimates of ∆u and |∇u|, we apply the maximum principle to the quantity of Hamilton's Laplacian estimate on complete noncompact Riemannian manifolds due to Karp-Ni [5] or Ni-Tam [9]. We remark that a priori integral bound needed for the application of maximum principle on complete noncompact manifolds has also been obtained in [3] and in [13] for more general setting.
As an application of Theorem 1.2, we obtain the following Laplacian estimate of the heat kernel on a complete noncompact Riemannian manifold with nonnegative Ricci curvature. Theorem 1.3. Let (M, g) be an n-dimensional complete noncompact Riemannian manifold with nonnegative Ricci curvature, and H(x, y, t) its heat kernel. Then, for all δ > 0, there exists a constant C = C(n, δ) such that for all x, y ∈ M and t > 0.
Remark 1.4. We would like to point out that Theorem 1.3 is sharp in the order of t for the heat kernel on R n .
The structure of this paper is organized as follows. In Section 2, we derive Bernstein-type gradient estimates of the Laplacian for solutions to the heat equation (see Theorem 2.7). Our proof makes use of Shi's gradient estimates [11], combining the classical cut-off function arguments. In Section 3, we finish the proof of Theorem 1.2 by using Theorem 2.7. In Section 4, we apply Theorem 1.2 to the heat kernel and complete the proof of Theorem 1.3.

Bernstein-type estimates
In this section, we assume that (M, g) be an n-dimensional complete noncompact Riemannian manifold with the Ricci curvature uniformly bounded below by −K for some constant K ≥ 0, and suppose that u is a smooth solution to the heat At first we recall the Kotschwar's result in [6].
for some p ∈ M n and A, R, T > 0. Then there exists a constant C = C(n, K) such that Remark 2.2. If Ric ≥ 0, from the proof course of Theorem 2.1 in [6], one shows that Remark 2.3. Letting R → ∞ in the proof course of Theorem 2.1, one immediately shows that there exists a constant C(n) such that In the above description, Kotschwar showed the first derivative estimate of the positive solution to the heat equation on complete manifolds. Below we will give an upper estimate of ∆u. Our proof is similar in spirit to the derivative estimates due to Shi [11] (see also [6]). Let where the constant C is to be chosen. The following lemma is useful for proving Theorem 2.7. . Also we can choose c := C −1 (n) · C −2 * . Moreover, if R → ∞, then lim R→∞ c is a finite positive constant. Note that here the constant C(n, K) may be different from the one in Theorem 2.1.
Remark 2.6. When K = 0, from (2.3), we see that the assumption T ≤ 1 can be replaced by T < ∞. In this case, from Remark 2.2, one can choose C * := C(n)A 2 1 + T R 2 and c := C −1 (n) · C −2 * . If R → ∞, then lim R→∞ c still be a finite positive constant, independent on T .
Using Lemma 2.4, we prove the following Laplacian estimate for the positive solution to the heat equation.
Therefore at (x 0 , t 0 ), we have Since c := C −1 (n) · C −2 * , we have that for any (x, t) ∈ B p (R) × [0, T ]. By the definitions of C * and G, we have and therefore for any (x, t) ∈ B p (R) × [0, T ], which completes the proof of the theorem.

Proof of Theorem 1.2
In this section by using gradient and Laplacian estimates of the previous section, we apply a maximum principle on complete noncompact manifolds due originally to Karp and Li [5] (see also ), to finish the proof of Theorem 1.2.
Proof of Theorem 1.2. Following the Hamilton's proof [4] with a little modification. We define u ǫ = u + ǫ satisfying ǫ < u ǫ < A + ǫ and the function By [4], we have By the Hamilton's arguments, we easily have that ∂ ∂t − ∆ P ≤ 0 whenever P ≥ 0.
This fact can be obtained by the following three cases.

Proof of Theorem 1.3
The proof of Theorem 1.3 follows from that of Theorem 1 in [6] with little modification, but is included for completeness.

Acknowledgment
The author would like to express his gratitude to the referee for careful readings and many valuable suggestions.