Volume functional of compact $4$-manifolds with a prescribed boundary metric

TL;DR

This paper characterizes critical metrics of the volume functional on 4-manifolds with boundary, showing they are isometric to standard space forms under certain curvature conditions, and provides related integral curvature estimates.

Contribution

It establishes a rigidity result for critical volume metrics on 4-manifolds with boundary, linking curvature conditions to standard space forms.

Findings

Critical metrics are isometric to space forms under Weyl tensor conditions.

Provides an integral curvature estimate involving the Yamabe constant.

Achieves a rigidity theorem for volume-critical metrics in 4D.

Abstract

We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.