Deformations of the hemisphere that increase scalar curvature

Details Deformations of the hemisphere that increase scalar curvature Deformations of the hemisphere that increase scalar curvature

2010-04-19

arxiv.org/abs/1004.3088v3

Mathematics

Differential Geometry

Revised version, to appear in Invent. Math

Scientific paper

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.

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