Abstract
Let (M, g) be a complete Riemannian 3-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature \({R \geq - 6}\). Building on work of A. Neves and G. Tian and of the first-named author, we show that the leaves of the canonical foliation of (M, g) are the unique solutions of the isoperimetric problem for their area. The assumption \({R \geq -6}\) is necessary. This is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity.
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Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). O. Chodosh is supported by the Oswald Veblen Fund and the NSF grants No. 1638352 and 1811059. M. Eichmair is supported by the START-Project Y963-N35 of the Austrian Science Fund (FWF). O. Chodosh and M. Eichmair thank the Erwin Schrödinger Institute for its warm hospitality during the program Geometry and Relativity in the Summer of 2017. They also thank Professors Hubert Bray, Simon Brendle, Gerhard Huisken, Jan Metzger, Richard Schoen, and Paul Yang for their kind support and encouragement. Y. Shi and J. Zhu are supported by the NSFC grants No. 11671015 and 11731001. They would like thank Professor Li Yuxiang for his kind support.
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Chodosh, O., Eichmair, M., Shi, Y. et al. Characterization of Large Isoperimetric Regions in Asymptotically Hyperbolic Initial Data. Commun. Math. Phys. 368, 777–798 (2019). https://doi.org/10.1007/s00220-019-03354-2
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DOI: https://doi.org/10.1007/s00220-019-03354-2