PRESCRIBING CURVATURES ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS WITH BOUNDARIES

Let (M, g) be a complete three dimensional Riemannian manifold with boundary. M. Given smooth functions K(x) > 0 and c(x) defined on M and partial derivative M, respectively, it is natural to ask whether there exist metrics conformal to g so that under these new metrics, K is the scalar curvature and c is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on K, c and (M, g) we show that all the solutions of the equation can only blow up at finite points over each compact subset of (M) over bar; some of them may appear on partial derivative M. We describe the asymptotic behavior of the blow-up solutions around each blow-up point and derive an energy estimate as a consequence.