An invariant related to the existence of conformally compact Einstein fillings

TL;DR

This paper introduces a new invariant for certain spin manifolds that indicates whether their boundary conformal class can extend to a conformally compact Einstein metric inside.

Contribution

It defines a novel invariant for compact spin manifolds with positive Yamabe boundary, linking its vanishing to the existence of conformally compact Einstein fillings.

Findings

Invariant vanishes if and only if a conformally compact Einstein filling exists

Provides a necessary condition for boundary conformal classes to admit Einstein fillings

Connects geometric analysis with conformal geometry in higher dimensions

Abstract

We define an invariant for compact spin manifolds $X$ of dimension $4k$ equipped with a metric $h$ of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of $h$ to be the conformal infinity of a conformally compact Einstein metric on $X$.