臺灣博碩士論文加值系統

在這篇報告裡, 我們探討在黎曼流型上多項式成長的調和函數理論, 其中主要內容著重於在非負Ricci曲率的完備黎曼流型上, 上述空間維度的估計問題. 這個問題源於1975年, 當S.T.Yau證明了在非負Ricci曲率的完備黎曼流型上, 所有非負的調和函數必為常數函數. Yau提出了一個猜想, 他認為在這種非負Ricci曲率的完備黎曼流型上, 多項式成長的調和函數空間為有限維度. 在一系列的論文中, Colding和Minicozzi證明了該猜想, 而且他們的結果更超越了Yau的猜想. 在這之後, P.Li和J.P.Wang做出了更進一步的結果.

In this report, we study the harmonic functions of polynomial growth on Riemannian manifolds, especially focus on estimate of the dimension of such spaces on complete Riemannian manifolds with non-negative Ricci curvature. This subject began in 1975, when S. T. Yau proved the strong Liouville property on complete manifolds with non-negative Ricci curvature. Yau conjectured that the space of harmonic functions of polynomial growth on Riemannian manifolds with non-negative Ricci curvature is finite dimensional. In a series of paper, Colding and Minicozzi proved and went beyond Yau's conjecture. P. Li and J. P. Wang proved further results on this topic afterwards.

1 Introduction 1
1.1 Historical background ............ 1
1.2 Linear growth .................... 8
1.3 Polynomial growth ................ 11
2 Notation and Some Related Properties 15
3 Harmonic Functions of Polynomial Growth 18
4 Example 29

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