No abstract

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let (M n , g 0 ) be a n-dimensional smooth compact manifold with boundary, where n ≥ 3, assume the conformal invariant Y (M, ∂M ) < 0. Given any negative smooth functions f in M and h on ∂M , there exists a unique conformal metric of g 0 such that its scalar curvature equals f and mean curvature curvature equals h. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In the flow approach, assume another conformal invariant Q(M, ∂M ) is a negative real number, for some class of initial data, we prove the short time and long time existences of the so-called prescribed scalar curvature plus mean curvature flows, as well as their asymptotic convergence. Via a family of such flows together with some additional variational arguments, under the flow assumptions we prove existence and uniqueness of positive minimizers of the associated energy functional and also the above result by analyzing asymptotic limits of the flows and the relations among some conformal invariants.

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension n ≥ 3. We prove the existence of such conformal metrics in the cases of n = 6, 7 or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be 1, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to +∞.