A Jang Equation Approach to the Penrose Inequality

TL;DR

This paper introduces a generalized Jang equation approach to address the Penrose Inequality in asymptotically flat spacetimes, providing new proofs in symmetric cases and outlining challenges for the general case.

Contribution

It develops a generalized Jang equation framework and proves existence and regularity results for spherically symmetric data, advancing the approach to the Penrose Inequality.

Findings

Established existence and regularity for spherically symmetric cases.

Provided a new proof of the Penrose Inequality for symmetric data.

Outlined the open problem of extending results to nonspherical cases.

Abstract

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 dominat 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.