Proof of the bounded conformal conjecture

TL;DR

This paper proves the bounded conformal conjecture for asymptotically flat 3-manifolds, establishing the existence of a conformally related metric with controlled minimal surface area, which has implications for geometric inequalities in general relativity.

Contribution

It proves the conformal conjecture assuming only the boundedness of the harmonic function, removing previous assumptions about finiteness of minimal enclosures.

Findings

Conformal conjecture holds under bounded harmonic function assumption.

Supports the use of conformal deformations in geometric inequalities.

Advances understanding of minimal surface behavior in asymptotically flat manifolds.

Abstract

Given any asymptotically flat 3-manifold $(M,g)$ with smooth, non-empty, compact boundary $\Sigma$, the conformal conjecture states that for every $\delta>0$, there exists a metric $g' = u^4 g$, with $u$ a harmonic function, such that the area of outermost minimal area enclosure $\tilde{\Sigma}_{g'}$ of $\Sigma$ with respect to $g'$ is less than $\delta$. 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 $\Sigma$, and boundedness of harmonic function $u$. We prove the conjecture assuming only the boundedness of $u$.