Abstract

A theorem of Escobar asserts that if a three-dimensional smooth compact Riemannian manifold M with boundary is of positive type and is not conformally equivalent to the standard three-dimensional ball, a necessary and suficient condition for a C² function H on M to be the mean curvature of some conformal scalar flat metric is that H be 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 C² norm and the total degree of all solutions is −1.

Mathematical Subject Classification

Primary: 35J60, 53C21, 58G30

Authors