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

Abstract

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。

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).