Critical metrics on $4$-manifolds with harmonic anti-self dual Weyl tensor

TL;DR

This paper characterizes certain 4-manifolds with harmonic anti-self dual Weyl tensor as geodesic balls in standard space forms, under specific boundary and critical metric conditions.

Contribution

It proves a rigidity theorem for 4-manifolds with harmonic anti-self dual Weyl tensor, identifying them as geodesic balls in space forms under boundary conditions.

Findings

Such manifolds are isometric to geodesic balls in space forms

The boundary being isometric to a standard sphere is crucial

The result applies to simply connected, compact critical metrics

Abstract

We prove that a $4$-dimensional simply connected, compact critical metric of the volume functional with harmonic anti-self dual Weyl tensor and boundary isometric to a standard sphere $\mathbb{S}^{3}$ is isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$