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Deformations of the hemisphere that increase scalar curvature

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA, 94305, USA

    Simon Brendle

  2. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

    Fernando C. Marques

  3. Imperial College, Huxley Building, 180 Queen’s Gate, London, SW7 2RH, UK

    Andre Neves

Authors
  1. Simon Brendle
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  2. Fernando C. Marques
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  3. Andre Neves
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Corresponding author

Correspondence to Simon Brendle.

Additional information

The first author was supported in part by the National Science Foundation under grant DMS-0905628. The second author was supported by CNPq-Brazil, FAPERJ, and the Stanford Department of Mathematics.

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Brendle, S., Marques, F.C. & Neves, A. Deformations of the hemisphere that increase scalar curvature. Invent. math. 185, 175–197 (2011). https://doi.org/10.1007/s00222-010-0305-4

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  • DOI: https://doi.org/10.1007/s00222-010-0305-4

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