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

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