在論文中，當Ricci曲率有下界，我們研究了完備黎曼流形上p-Laplacian的譜幾何和梯度估計的性質。首先，我們證明了拉普拉斯算子的分裂型定理。然後，我們給出與p-Laplacian的第一特徵值對應的正p-特徵函數的局部梯度估計。此外，我們證明了p-特徵函數的最優全局梯度估計。另一方面，我們首先推導出正的p-Laplacian熱方程解的Li-Yau型梯度估計。最後，我們證明了在一些不同的曲率假設下，p-Laplacian的第一個特徵值沿著Ricci-Bourguignon流幾乎是隨處嚴格單調遞增和可微分的。

In this thesis, we study the properties of the spectrum geometric and gradient estimates for p-Laplacian on complete Riemannian manifolds with the Ricci curvature is bounded from below. First, we prove a splitting type theorem for the Laplacian. Then, we give a local gradient estimate for the positive p-eigenfunction associated to the first eigenvalue of the p-Laplacian. Moreover, we show global sharp gradient estimates for p-eigenfunctions. On the other hand, we first derive a Li-Yau type gradient estimate for the positive solutions to the p-Laplacian heat equation. Finally, we prove that the first eigenvalue of the p-Laplacian is strictly monotone increasing and differentiable almost everywhere along with the RicciBourguignon flow under some different curvature assumptions.

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