Critical Metrics and Curvature of Metrics with Unit Volume or Unit Area of the Boundary

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