  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 $C^2$ 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.
