# Poincar\'e-Einstein metrics and Yamabe invariants

## TL;DR

This paper demonstrates the existence of many conformal classes on S^7 that do not correspond to Poincaré-Einstein metrics on B^8 and establishes a precise inequality relating Yamabe invariants of boundary and interior.

## Contribution

It proves the existence of infinitely many conformal classes on S^7 not realizable as Poincaré-Einstein boundaries and establishes a sharp Yamabe invariant inequality.

## Key findings

- Infinitely many conformal classes on S^7 cannot be boundary of Poincaré-Einstein metrics.
- A sharp inequality between boundary and interior Yamabe invariants is established.
- The results deepen understanding of conformal geometry and Poincaré-Einstein metrics.

## Abstract

In this note we prove the existence of infinitely many positive conformal classes on $S^7$ which cannot be the conformal infinity of a Poincar\'e-Einstein metric on the ball $B^8$. We also prove a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior (after a suitable compactification).