(M,g)是一個Ricci曲率有下界完備n維黎曼流形。在這篇文章中，我們研究在這種流形上熱方程的漢彌爾頓梯度估計。

Let (M,g) be a complete n-dimensional Riemannian manifold with Ricci curvature bounded from blew. In this note, we study the Hamilton’s gradient estimate for the heat equations on such manifolds.

1. Introduction - 2
2. Preliminary - 3
2.1. Gradient estimate and Harnack inequality - 7
2.2. Li-Yau type of gradient estimate and Harnack inequality on heat equation - 13
2.3. Upper and lower bound of heat Kernel - 20
3. Main Theorem - 27
References - 33

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