Deformations of the hemisphere that increase scalar curvature

TL;DR

This paper constructs counterexamples to Min-Oo's conjecture, which posited that certain scalar curvature conditions on a manifold with a spherical boundary imply it is isometric to a standard hemisphere.

Contribution

The paper provides the first known counterexamples to Min-Oo's conjecture in dimensions three and higher, challenging previous assumptions about scalar curvature rigidity.

Findings

Counterexamples exist in dimensions n ≥ 3

Min-Oo's conjecture does not hold universally

Implications for scalar curvature rigidity theories

Abstract

Consider a compact Riemannian manifold M of dimension n whose boundary \partial M is totally geodesic and is isometric to the standard sphere S^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n-1), then M is isometric to the hemisphere S_+^n equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases. In this paper, we construct counterexamples to Min-Oo's conjecture in dimension n \geq 3.