Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let ((M,g_0)) be a smooth compact manifold of dimension (nge 3) with boundary. Given any smooth functions f in M and h on (partial M), does there exist a conformal metric of (g_0) such that its scalar curvature equals f and boundary mean curvature equals h? Assume that f and h are negative and the conformal invariant (Q(M,partial M)) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.