A remark on the rigidity of conformally compact Poincar{\'e}-Einstein manifolds

TL;DR

This paper establishes an optimal inequality linking the Yamabe invariants of conformally compact Poincaré-Einstein manifolds and their boundaries, providing a new proof of hyperbolic space rigidity.

Contribution

It introduces an optimal inequality relating Yamabe invariants and offers an elementary proof of hyperbolic space rigidity.

Findings

Derived an optimal inequality for Yamabe invariants.

Proved hyperbolic space is uniquely rigid among conformally compact Poincaré-Einstein manifolds.

Provided an elementary proof of the rigidity result.

Abstract

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{\'e}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the only conformally compact Poincar{\'e}-Einstein manifold with the round sphere as its conformal infinity.