Prescribing curvatures on three dimensional Riemannian manifolds with boundaries - Mathematics > Differential Geometry

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Abstract: Let $M,g$ be a complete three dimensional Riemannian manifold with boundary$\partial M$. Given smooth functions $Kx>0$ and $cx$ defined on $M$ and$\partial M$, respectively, it is natural to ask whether there exist metricsconformal to $g$ so that under these new metrics, $K$ is the scalar curvatureand $c$ is the boundary mean curvature. All such metrics can be described by aprescribing curvature equation with a boundary condition. With suitableassumptions on $K$,$c$ and $M,g$ we show that all the solutions of theequation can only blow up at finite points over each compact subset of $\barM$, some of them may appear on $\partial M$. We describe the asymptoticbehavior of the blowup solutions around each blowup point and derive an energyestimate as a consequence.

Autor: Lei Zhang

Fuente: https://arxiv.org/