Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

TL;DR

This paper establishes new rigidity results for 4-dimensional compact manifolds with harmonic Weyl tensor under biorthogonal curvature bounds, improving previous pinching constant conditions.

Contribution

It introduces a rigidity theorem based on biorthogonal curvature, a weaker condition than sectional curvature, enhancing existing pinching results for 4-manifolds.

Findings

Rigidity under biorthogonal curvature bounds

Improved pinching constants for 4-manifold rigidity

Extension of previous curvature pinching results

Abstract

The goal of this paper is to investigate the rigidity of 4-dimensional manifolds involving some pinching curvature conditions. To this end, we make use of the approach of biorthogonal curvature which is weaker than the sectional curvature. Here, we prove a rigidity result for 4-dimensional compact manifolds under a suitable lower bound condition on the minimum of the biorthogonal curvature. From this, we improve the pinching constants considered by some preceding works on a rigidity result for 4-dimensional manifolds.