A Jang equation approach to the Penrose inequality

Abstract

We introduce a generalized version of the Jang equation, designedfor the general case of the Penrose Inequality in the setting ofan asymptotically flat space-like hypersurface of a spacetimesatisfying the dominant energy condition. The appropriateexistence and regularity results are established in the specialcase of spherically symmetric Cauchy data, and are applied to givea 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 associatedsystem of equations in the nonspherical setting would yield aproof of the full Penrose Conjecture. Thus it remains as animportant and challenging open problem to determine whether thissystem does indeed admit the desired solutions.