Deformations of the hemisphere that increase scalar curvature

Understand

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.