Critical metrics and curvature of metrics with unit volume or unit area of the boundary

TL;DR

This paper investigates the variational properties of volume and boundary area functionals on manifolds with boundary, characterizing critical metrics and solving curvature problems using the Yamabe invariant.

Contribution

It provides necessary and sufficient conditions for critical metrics with prescribed curvature and introduces an analogue of V-statics metrics, solving boundary curvature problems.

Findings

Characterization of critical metrics for volume and area functionals.

Solution to the Kazdan-Warner-Kobayashi problem with boundary constraints.

Conditions for prescribed scalar and mean curvatures based on the Yamabe invariant.

Abstract

Given a smooth compact manifold with boundary, we study variational properties of the volume functional and of the area functional of the boundary, restricted to the space of the Riemannian metrics with prescribed curvature. We obtain a sufficient and necessary condition for a metric to be a critical point. As a by-product, a very natural analogue of V-statics metrics is obtained. In the second part, using the Yamabe invariant in the boundary setting, we solve the Kazdan-Warner-Kobayashi problem in a compact manifold with boundary. For several cases, depending on the signal of the Yamabe invariant, we give sufficient and necessary condition for a smooth function to be the scalar or mean curvature of a metric with constraint on the volume or area of the boundary.