Inventiones mathematicae

, Volume 185, Issue 1, pp 175–197 | Cite as

Deformations of the hemisphere that increase scalar curvature

  • Simon Brendle
  • Fernando C. Marques
  • Andre Neves


Consider a compact Riemannian manifold M of dimension n whose boundary ∂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≥3.


Scalar Curvature Dirac Operator Hyperbolic Manifold Compact Riemannian Manifold Riemannian Metrics 
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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Simon Brendle
    • 1
  • Fernando C. Marques
    • 2
  • Andre Neves
    • 3
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Instituto de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  3. 3.Imperial CollegeLondonUK

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