May  2010, 27(2): 741-766. doi: 10.3934/dcds.2010.27.741

A Jang equation approach to the Penrose inequality

1. 

Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, United States

2. 

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, United States

Received  October 2009 Revised  February 2010 Published  February 2010

We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominant energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data, and are applied to give a new proof of the general Penrose Inequality for these data sets. When appropriately coupled with an inverse mean curvature flow, analogous existence and regularity results for the associated system of equations in the nonspherical setting would yield a proof of the full Penrose Conjecture. Thus it remains as an important and challenging open problem to determine whether this system does indeed admit the desired solutions.
Citation: Hubert L. Bray, Marcus A. Khuri. A Jang equation approach to the Penrose inequality. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 741-766. doi: 10.3934/dcds.2010.27.741
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