Resumen en español

Let (Mⁿ, g) be an n – dimensional compact Riemannian manifold with boundary with n ≥ 2. In this paper we study the uniquensess of metrics in the conformal class of the metric g having the same scalar curvature in M, aM, and the same mean curvature on the boundary of M, aM. We prove the equivalence of some uniqueness results replacing the hypothesis on the first Neumann eigenvalue of a linear elliptic problem associated to the problem of conformal deformations of metrics for one about the first Dirichlet eigenvalue of that problem.