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On the Prescribed Scalar and Zero Mean Curvature on 3-Dimensional Manifolds With Umbilic Boundary

Abstract In this paper we consider the existence and the compactness of Riemannian metrics of prescribed mean curvature and zero boundary mean curvature on a three dimensional manifold with umbilic boundary (M, g0). We prove that for three dimensional manifolds with umbilic boundaries, which are not conformally equivalent to the three dimensional standard half sphere, any positive function can be realized as the scalar curvature of a Riemannian metric g conformal to g0 with respect to which the boundary has zero mean curvature. Moreover, all such metrics stay bounded with respect to the C2,α -topology and in the nondegerate case Morse inequalities hold.