Prescribing curvatures on three dimensional Riemannian manifolds with boundaries

Zhang, Lei

doi:10.1090/S0002-9947-09-04911-3

Prescribing curvatures on three dimensional Riemannian manifolds with boundaries
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Trans. Amer. Math. Soc. 361 (2009), 3463-3481 Request permission

Abstract:

Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, 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 $\bar M$; some of them may appear on $\partial M$. We describe the asymptotic behavior of the blow-up solutions around each blow-up point and derive an energy estimate as a consequence.

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