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Rigidity for critical metrics of the volume functional
Author(s) -
Barros A.,
da Silva A.
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700240
Subject(s) - mathematics , scalar curvature , rigidity (electromagnetism) , ricci curvature , geodesic , conjecture , simply connected space , boundary (topology) , bounded function , manifold (fluid mechanics) , pure mathematics , constant curvature , curvature , mathematical analysis , geometry , mechanical engineering , engineering , structural engineering
Geodesic balls in a simply connected space forms S n , R n or H n 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. [7][W. Boucher, 1984] and Shen [18][Y. Shen, 1997] to n ‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.