完備流形上的調和映射

A note on harmonic maps

鄭穎鍾(Ying-Chung Cheng)

指導教授 ： 宋瓊珠

國立清華大學/理學院/數學系/碩士(2014年)

https://doi.org/10.6843/NTHU.2014.00335

摘要

此篇論文中含有兩個主題，一是p-調和映射的凸集合叢簇性質，另一是在有Weighted Poincare'不等式的完備流形上的Lioville定理。
1.p-調和映射的凸集合叢簇性質：
在這篇論文中，我們介紹了從完備流形到Cartan-Hadamard的p-調和映射，並且估計其值域。我們也給出了相異massive集合最大數量的上界。

2.在有Weighted Poincare'不等式的完備流形上的Lioville定理：
令M是一個有下界Ricci曲率的完備非緊緻流形，N是一個有非正sectional曲率的完備流形。假設Weighted Poincare'不等式成立與Dirichlet能量函數有適當成長的話，我們證明了從M到N的調和映射之Liouville性質。

關鍵字

調和映射

並列摘要

There are two topics in this paper. One is Convex Hull Property of p-harmonic maps, and the other is Liouville theorems on manifolds with weighted poincare' inequality.
1.Convex Hull Property Of P-harmonic Maps：
In this paper, we introduce the p-harmonic maps on complete manifolds to Cartan-Hadamard manifolds and estimate the image of the maps. We give the upper bound for the maximum number of disjoint massive sets.

2.Liouville Theorems On Manifolds With Weighted Poincare' Inequality：

Let M be a complete noncompact manifold with some Ricci curvature lower bound and N be a complete manifold with nonpositive sectional curvature. We prove the Liouville property on harmonic maps from M to N provided that the weighted Poincare' inequality holds and the Dirichlet energy function of the harmonic map has a proper growth.

並列關鍵字

harmonic maps

參考文獻

1.Convex Hull Property Of P-harmonic Maps：

G : A. Grigor'yan, Dimensions of spaces of harmonic functions, Matem. Zametki. 48 (1990), 55-61.

H : B.Hua, Generalized Liouville theorem in nonnegatively curved Alexandrov spaces, Chin. Ann. Math. Ser. B 30 (2009), no. 2, 111-128.

H-L : Han, Q. and Lin, F. H., Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, Vol. 1, A. M. S., Providence, RI, 1997.

J : Anand Arvind Joshi, Harmonic mappings between Riemannian manifolds. (2006)