在本論文中，我們首先給出了拉普拉斯特徵函數梯度估計的完整證明，然後給出了p-特徵函數的銳梯度估計。最後，我們詳細證明了主特徵值達到最大值的流形結構。

In this thesis, we first give a complete proof of a gradient estimate for positive eigenfunctions of Laplacian, then we show a sharp gradient estimate for positive p-eigenfunctions. At last, we give a detailed proof of the theorem of Sung and Wang in the structure of manifolds whose principal eigenvalues achieve the maximum value.

1 Introduction---------------------------1
2 Preliminaries--------------------------4
3 Gradient estimates for Laplacian-------11
4 Gradient estimates for p-Laplacian-----21
5 Splitting theorems---------------------48

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