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The equality case of the Penrose inequality for asymptotically flat graphs


Authors: Lan-Hsuan Huang and Damin Wu
Journal: Trans. Amer. Math. Soc. 367 (2015), 31-47
MSC (2010): Primary 53C24; Secondary 83C99
DOI: https://doi.org/10.1090/S0002-9947-2014-06090-X
Published electronically: September 18, 2014
MathSciNet review: 3271252
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Abstract: We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we developed in our paper, Hypersurfaces with non-negative scalar curvature (J. Differential Geom., vol. 95 (2013), pp. 249-278). This enables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation.


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Additional Information

Lan-Hsuan Huang
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: lan-hsuan.huang@uconn.edu

Damin Wu
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: damin.wu@uconn.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06090-X
Received by editor(s): May 11, 2012
Published electronically: September 18, 2014
Additional Notes: The first author acknowledges NSF grant DMS-$1005560$ and DMS-$1301645$ for partial support.
Article copyright: © Copyright 2014 American Mathematical Society