P. D. E.'s Which Imply the Penrose Conjecture

Abstract

Abstract. In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool in our method is the derivation of a new identity which we call the generalized Schoen-Yau identity, which is of independent interest. Using a generalized Jang equation, we use this identity to propose canonical embeddings of Cauchy data into corresponding static spacetimes. In addition, we discuss the Carrasco-Mars counterexample to the Penrose conjecture for generalized apparent horizons (added since the first version of this paper was posted on the arXiv) and instead conjecture the Penrose inequality for time-independent apparent horizons, which we define.

Transformations of asymptotically AdS hyperbolic initial data and associated geometric inequalities

“…The existence, regularity, and blow-up behavior for solutions of the generalized Jang equation was studied in [28], and discussed in [7,8]. Moreover, in [14] it was shown that the new data possess the desired asymptotics and that the mass is preserved up to a contribution from the warping factor.…”

“…The purpose of the generalized Jang equation is to impart a desired lower bound for the scalar curvature of g 1 . Namely, as is shown in [7,8] we have…”

“…Consider an inverse mean curvature flow (IMCF) { S τ } inside (M 3 , g 3 ) emanating from the minimal boundary S 0 = ∂M 3 . According to the arguments of [7,8,14] together with (4.11),…”

“…The deformation procedures consist of three parts. First, similar to [7,8], the given asymptotically AdS hyperbolic data (M, g, k) is transformed into a time symmetric data set (M 1 , g 1 ) which is also asymptotically AdS hyperbolic, has the same mass, and satisfies the scalar curvature lower bound R 1 ≥ −6 weakly. Second, following [21], (M 1 , g 1 ) is transformed into an umbilic asymptotically hyperboloidal data set (M 2 , g 2 , k 2 ) which again preserves the mass and satisfies R 2 ≥ −6 weakly.…”

“…In [14] we proposed deformations of asymptotically hyperboloidal initial data which transformed the asymptotically hyperbolic structure into an asymptotically flat structure, while preserving the mass up to scaling by a positive constant. These deformations are based on solutions of the so called Jang-type equations [7,8,12,13,33,46], which impart positivity properties to the scalar curvature of the deformed data. In effect, the full round of geometric inequalities in the asymptotically hyperboloidal setting is reduced to their counterparts in the asymptotically flat setting modulo the solution of canonical system of partial differential equations.…”