Self-dual and even Poincar\'e-Einstein metrics in dimension four

TL;DR

This paper establishes rigidity and gap theorems for self-dual and even Poincaré-Einstein metrics in four dimensions, providing new obstructions based on boundary invariants and topology, and introduces a novel scalar conformal invariant.

Contribution

It proves new rigidity and gap theorems for these metrics, links boundary invariants to existence obstructions, and identifies a new scalar conformal invariant in three dimensions.

Findings

Rigidity and gap theorems for four-dimensional self-dual Poincaré-Einstein metrics.

Obstructions to existence based on boundary conformal invariants.

Discovery of a new scalar conformal invariant in three-dimensional manifolds.

Abstract

We prove rigidity and gap theorems for self-dual and even Poincar\'e-Einstein metrics in dimension four. As a corollary, we give an obstruction to the existence of self-dual Poincar\'e-Einstein metrics in terms of conformal invariants of the boundary and the topology of the bulk. As a by-product of our proof we identify a new scalar conformal invariant of three-dimensional Riemannian manifolds.