Isoperimetric inequality and Weitzenb\"ock type formula for critical metrics of the volume

TL;DR

This paper establishes an isoperimetric inequality and a Weitzenb"ock type formula for critical volume metrics with nonnegative scalar curvature, leading to classification results on compact manifolds with boundary.

Contribution

It introduces new inequalities and formulas for critical volume metrics, extending understanding of their geometric properties and classifications.

Findings

Proves an isoperimetric inequality for critical volume metrics.

Derives a Weitzenb"ock type formula in four dimensions.

Classifies critical metrics under certain conditions.

Abstract

We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenb\"ock type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a classification result for such metrics.