A geometric equation with critical nonlinearity on the boundary

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 sufficient condition for a C 2 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 C 2 norm and the total degree of all solutions is -1.