Proof of the bounded conformal conjecture

Abstract

Given any asymptotically flat 3-manifold $(M,g)$ with smooth, non-empty, compact boundary $\hat{I}\pounds$, the conformal conjecture states that for every $\hat{I}´>0$, there exists a metric $g^{\prime }=u^{4}g$, with $u$ a harmonic function, such that the area of outermost minimal area enclosure $\tilde{\hat{I}}{\pounds }_{g^{\prime }}$ of $\hat{I}\pounds$ with respect to $g^{\prime }$ is less than $\hat{I}´$. Recently, the conjecture was used to prove the Riemannian Penrose inequality for black holes with zero horizon area, and was proven to be true under the assumption of existence of only a finite number of minimal area enclosures of boundary $\hat{I}\pounds$, and boundedness of harmonic function $u$. We prove the conjecture assuming only the boundedness of $u$.

Few changes to the proof of Proposition 1. Result unchanged, 25 pages