完備流行上的結構定理

Some Splitting and Vanishing Type Theorems on Complete Manifolds

阮碩勇(Nguyen Thac Dung)

指導教授 ： 宋瓊珠

國立清華大學/理學院/數學系/博士(2012年)

摘要

In this Dissertation, we study several splitting and vanishing type theorems on some Riemannian manifolds by studying the classes of harmonic functions and of harmonic $p$-forms. First, we prove splitting and vanishing theorem on complete Riemannian manifolds of dimension $ngeq3$ with the Ricci curvature is bounded from below in terms of bottom of spectrum of Laplacian. Moreover, we consider the Riemannian manifolds with weighted Poincar'{e} type inequality. Assuming a growth condition on the weight function, several splitting theorems and vanishing theorems are proved, moreover some vanishing properties of a special class of $p$-forms are also given on these such spaces. We also consider the smooth metric measure space with Bakry-'{E}mery curvature. We show a splitting property of smooth metric measure space with the Bakry-'{E}mery curvature bounded from below in terms of bottom spectrum of weighted Laplacian. Then, we consider a smooth metric measure space of infinite dimension Bakry-'{E}mery curvature with a weighted Poincar'{e} inequality. We are successfully in showing very general splitting results in this space. Many works done before now are reproved by them. Last, we mention a special smooth metric measure space, the gradient Ricci soliton. Vanishing type theorems of the class of harmonic $1$-forms which has finite $L^p$-norm are given.

關鍵字

Bakry-Emery曲率 ； 調和p-型 ； Riemannian流形 ； 光滑可測度空間

並列摘要

無資料

並列關鍵字

Bakry-Emery curvature ； Harmonic p-forms ； Riemannian manifolds ； Smooth metric measure spaces

參考文獻

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