On the scalar curvature rigidity for mainifolds with non-positive Yamabe invariant

TL;DR

This paper establishes a scalar curvature rigidity result for manifolds with non-positive Yamabe invariant, showing that under certain curvature conditions, the manifold must be Einstein, extending previous results to more general cases.

Contribution

The paper generalizes scalar curvature rigidity results to manifolds with non-positive Yamabe invariant, using analysis of geometric evolution equations.

Findings

Manifolds with scalar curvature not less than the Yamabe invariant are isometric to Einstein manifolds.

Extension of previous rigidity theorems from zero to non-positive Yamabe invariant cases.

Dependence on geometric evolution equations for establishing rigidity.

Abstract

In this paper, we study scalar curvature rigidity of non-smooth metrics on smooth manifolds with non-positive Yamabe invariant. We prove that if the scalar curvature is not less than the Yamabe invariant in distributional sense, then the manifold must be isometric to an Einstein manifold. This result extends Theorem 1.4 in Jiang, Sheng and the first author (Sci. China Math. 66 (2023) no. 6, 1141-1160), from a special case where the manifolds have zero Yamabe invariant to general cases where the manifolds have non-positive Yamabe invariant. This result depends highly on an analysis and estimates of geometric evolution equations.