Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds

TL;DR

This paper provides an elementary, self-contained approach to establishing the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities, using Fredholm theory on asymptotically hyperbolic manifolds.

Contribution

It introduces a simplified derivation of Fredholm theorems for geometric elliptic operators, facilitating the construction of Einstein metrics with specific boundary conditions.

Findings

Existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities.

Elementary derivation of Fredholm theorems for geometric operators.

Application to metrics close to a given Einstein metric with nonpositive curvature.

Abstract

The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an elementary derivation of sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds.