Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61225This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the $C^2$-norm and the total Leray-Schauder degree of all solutions is equal to -1. Then we deduce from this compactness result the existence of at least one solution to our problem.34 pagesAnalysis of PDEs35J60; 53C21; 58G30Compactness results in conformal deformations of Riemannian metrics on manifolds with boundariestext