Four manifolds with postive Yamabe constant

TL;DR

This paper refines a key theorem on four-manifolds with positive Yamabe constant, providing sharp rigidity and classification results based on conformal invariants, extending previous sphere theorems.

Contribution

It offers new sharp rigidity and classification theorems for four-manifolds with positive Yamabe constant, generalizing existing conformally invariant sphere theorems.

Findings

Sharp rigidity theorems for four-manifolds with positive Yamabe constant.

Classification results based on conformal invariants.

Extension of the conformally invariant sphere theorem.

Abstract

We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality. We prove some classification theorems of four manifolds according to some conformal invariants (see Theorems 1.3 and 1.6), which generalize the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}.