我們先介紹研究調和函數的基本工具梯度估計和 Harnack 不等式。接下來應用龐加萊和均值不等式來研究調和函數和半調和函數在黎曼流形的性質。在第三章我們證明 L^{1} 函數在拉普拉斯方程式和熱方程式上的解具有唯一性。最後，我們假設黎曼流形具有非負的瑞奇曲率和 maximal volume growth , 然後討論格林函數的行為和估計 heat kernel 。

We introduce Gradient Estimate and Harnack Inequality , that are essential to the study of harmonic functions . Next we apply Poincare and mean value inequality to study some properties of
harmonic and subharmonic functions on Riemannian manifold . In Chapter 3 , we show the uniqueness properties for L^{1} solutions of the Laplace equation and the heat kernel equation .
Finally , assume that manifold M with nonnegative Ricci curvature and maximal volume growth , then we discuss the behavior of the Green function and estimate the heat kernel .

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