Some geometric equations with critical nonlinearity on the boundary

A 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 C2 function H to be the mean curvature of some conformal flat metric is that H is positive somewhere. We show that all such metrics stay in a compact set with respect to the C2 norm and the total degree of all solutions is equal to −1. Similar existence and compactness results are also obtained for more general equations. MSC classification: 35J60, 53C21, 58G30.