The scalar curvature and the biorthogonal curvature: A pinching problem

TL;DR

This paper investigates a scalar curvature pinching condition in four-dimensional manifolds, extending classical sectional curvature results to scalar curvature, and provides new insights into the topology of such manifolds.

Contribution

It offers a novel scalar curvature pinching theorem in dimension four, addressing a question posed by S. T. Yau about replacing sectional curvature with scalar curvature.

Findings

Established a scalar curvature pinching condition implying the manifold is homeomorphic to a sphere.

Extended classical curvature pinching results to scalar curvature in four dimensions.

Provided a partial answer to Yau's problem in the context of scalar curvature.

Abstract

The famous pinching problem says that on a compact simply connected $n$-manifold if its sectional curvature satisfies $K_{min} > (1/4)K_{max} > 0$, then the manifold is homeomorphic to the sphere. In [8, problem 12], S. T. Yau proposed the following problem: If we replace $K_{max}$ by the scalar curvature, can we deduce similar pinching theorems? In our present note we give an answer to this question in dimension $n = 4.$}