A Local Existence Result for Poincar\'e-Einstein metrics

TL;DR

This paper proves the local existence of conformally compact Einstein metrics near a given closed Riemannian manifold, establishing a foundational result in geometric analysis and Einstein geometry.

Contribution

It introduces a new local existence theorem for Poincaré-Einstein metrics with prescribed conformal infinity on a collar neighborhood.

Findings

Existence of conformally compact Einstein metrics near a given boundary metric.

Construction of Einstein metrics with specified conformal infinity.

Extension of local geometric analysis techniques to Einstein metrics.

Abstract

Given a closed Riemannian manifold $(M, g_M)$ of dimension $n \geq 3$, we prove the existence of a conformally compact Einstein metric $g_{+}$ defined on a collar neighborhood $M \times (0,1]$ whose conformal infinity is $[g_M]$.