2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61732A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian manifold with boundary which is not conformally equivalent to the standard three dimensional ball, a necessary and sufficient condition for a $C^2$ function $H$ to be the mean curvature of some conformal flat metric is that $H$ is positive somewhere. We show that, when the boundary is umbilic and the function $H$ is positive everywhere, all such metrics stay in a compact set with respect to the $C^2$ norm and the total degree of all solutions is equal to -1.28 pagesAnalysis of PDEsDifferential Geometry35J60, 53C21, 58G30text