Low regularity Poincar\'e-Einstein metrics

TL;DR

This paper constructs a specific low-regularity Einstein metric on the ball with particular asymptotic curvature decay, demonstrating the existence of such metrics with limited smoothness at the boundary.

Contribution

It proves the existence of a $C^{1,1}$ conformally compact Einstein metric with precise asymptotic decay properties, showing such metrics can lack higher regularity.

Findings

Existence of a $C^{1,1}$ Einstein metric on the ball.

The metric's sectional curvature decays to -1 with exponential correction.

The metric admits no $C^2$ conformal compactification.

Abstract

We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has no $C^2$ conformal compactification.