Uniqueness of solutions of the Yamabe problem on manifolds with boundary. (October 2019) - Cárdenas, Elkin; Sierra, Willy
Abstract: Given a compact manifold with boundary M of dimension m ≥ 3 and a nondegenerate Riemannian metric g ∗ having null scalar curvature, constant mean curvature, and unitary volume on the boundary, we show that the set of Riemannian metrics with null scalar curvature, constant mean curvature, and unitary volume on the boundary, near to g ∗ is an embedded submanifold of the manifold of all Riemannian metrics on M . Additionally, such submanifold is strongly transversal to the conformal classes. We also prove, using recent results of compactness, that conformal classes of metrics closed to g ∗ contain only one of these metrics. Highlights: We study results related to the Yamabe problem on compact manifolds with boundary M . We prove a result of global uniqueness for solutions of the Yamabe problem on M . A certain set of metrics is a submanifold of the set of Riemannian metrics on M .