Prescribing curvatures on three dimensional Riemannian manifolds with boundaries
Lei Zhang · Transactions of the American Mathematical Society · 2009
Let ( M , g ) (M,g) be a complete three dimensional Riemannian manifold with boundary ∂ M \partial M . Given smooth functions K ( x ) > 0 K(x)>0 and c ( x ) c(x) defined on M M and ∂ M \partial M , respectively, it is natural to ask whether there exist metrics conformal to g g so that under these new metrics, K K is the scalar curvature and c 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 K , c c and ( M , g ) (M,g) we show that all the solutions of the equation can only blow up at finite points over each compact subset of M ¯ \bar M ; some of them may appear on ∂ M \partial M . We describe the asymptotic behavior of the blow-up solutions around each blow-up point and derive an energy estimate as a consequence.