Differential geometry via harmonic functions

Abstract

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 $\lambda_1(M) >0$ and whose Ricci curvature is bounded by $$ Ric_M \ge -{m-2 \over m-1} \lambda_1(M). $$ 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.