arXiv math/03042642003-04-18math.DG

通过调和函数的Differential geometry

Differential geometry via harmonic functions

Peter Li

发表信息：: Proceedings of the ICM, Beijing 2002, vol. 2, 293--302

摘要：

在这次演讲中，我将讨论使用调和函数来研究完备流形的几何和拓扑。在我以前与Luen-fai Tam的合作工作中，我们发现，一个完备流形的无穷大个数可以通过调和函数空间的维数来估计。应用这一点到一个完备流形的里奇曲率几乎是非负的，我们证明了流形必须有许多端。在我最近与王嘉平的合作中，我们成功地将这个一般方法应用于另外两类完备流形。第一类是具有谱$λ_1(M) >0$下界且里奇曲率以$$ Ric_M \ge -{m-2 \over m-1} λ_1(M). $$为界的流形。第二类是完备流形中具有非负截面曲率的稳定极小超曲面。在这两种情况下，我们证明了一些分裂型定理和一些有限性定理。

英文摘要：

In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be estimated by the dimension of a certain space of harmonic functions. Applying this to a complete manifold whose Ricci curvature is almost non-negative, we showed that the manifold must have finitely many ends. In my recent joint works with Jiaping Wang, we successfully applied this general method to two other classes of complete manifolds. The first class are manifolds with the lower bound of the spectrum $λ_1(M) >0$ and whose Ricci curvature is bounded by $$ Ric_M \ge -{m-2 \over m-1} λ_1(M). $$ The second class are stable minimal hypersurfaces in a complete manifold with non-negative sectional curvature. In both cases we proved some splitting type theorems and also some finiteness theorems.

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摘要：

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