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Rigidity for critical metrics of the volume functional

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  • A. Barros
  • A. da Silva

Abstract

Geodesic balls in a simply connected space forms Sn, Rn or Hn are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. and Shen to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.

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  • A. Barros & A. da Silva, 2019. "Rigidity for critical metrics of the volume functional," Mathematische Nachrichten, Wiley Blackwell, vol. 292(4), pages 709-719, April.
    • Handle: RePEc:bla:mathna:v:292:y:2019:i:4:p:709-719
    DOI: 10.1002/mana.201700240
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    Cited by:

    1. Rafael Diógenes & Tiago Gadelha, 2022. "Compact quasi‐Einstein manifolds with boundary," Mathematische Nachrichten, Wiley Blackwell, vol. 295(9), pages 1690-1708, September.
    2. Huiya He, 2023. "Critical metrics of the volume functional on three‐dimensional manifolds," Mathematische Nachrichten, Wiley Blackwell, vol. 296(7), pages 2838-2849, July.

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