班克明-愛茉莉擬赫米遜里奇曲率下完備擬赫米遜流型上L-擬調和函數的梯度估計及劉維爾性質

Gradient Estimate and Liouville Property of L-pseudoharmonic Functions on a Complete Pseudohermitian Manifold with Bakry-Emery Pseudohermitian Ricci Curvature

游禮中(Li-Chung Yu)

指導教授 ： 張樹城

國立臺灣大學/理學院/數學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.02296

摘要

這篇論文主要是模仿丘成桐教授在1975年對黎曼流型上的L同調函數作梯度估計的方法，進一步的引用在加權柯西黎曼流型上面。由於考慮的為加權流型，因此考慮的曲率將從柯西黎曼里奇曲率，調整成考慮Bakry-Emery 里奇曲率。經由論文內的計算我們可得知，當這個曲率具有下界時，L同調正函數的梯度將會有上界。更進一步來說，若此曲率的下界為零時，則此函數將會是常數函數。換句話說，當此曲率下界為零，我們將會得到加權柯西黎曼流型上的劉維爾定理。

關鍵字

加權流型、柯西黎曼流型 ； Bakry-Emery 里奇曲率 ； L同調函數 ； 梯度估計

並列摘要

In this paper, we modify Yau's method to discuss a gradient estimate of a nonnegative L-pseudoharmonic function on a oriented, complete, pseudohermitian manifold which satisfies Witten-sub-Laplacian comparison property. Since the manifold we considered in this paper is weighted manifold, the curvature we consider is not only Ricci curvature but Bakry-Emery Ricci curvature Ric_m,n (L). At the end of this paper, we can get that when the form 2Ric_m,n (L) - Tor(L) is bounded below, any gradient estimate of a nonnegative L-pseudoharmonic function is bounded. Moreover, we can then deduce Liouville property on such manifold with curvature satisfies 2Ric_m,n (L) > Tor(L).

並列關鍵字

weighted manifold ； CR manifold ； Bakry-Emery Ricci curvature ； L-harmonicfunction ； gradient estimate

參考文獻

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