Prescribing curvatures on three dimensional Riemannian manifolds with boundaries

TL;DR

This paper investigates the existence and behavior of conformal metrics on three-dimensional Riemannian manifolds with boundary, prescribing scalar and boundary mean curvatures, and analyzes blowup phenomena of solutions.

Contribution

It establishes conditions under which solutions to the prescribing curvature problem blow up finitely and describes their asymptotic behavior near blowup points.

Findings

Solutions blow up only at finitely many points within the manifold.

Some blowup points may occur on the boundary.

An energy estimate for blowup solutions is derived.

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 blowup solutions around each blowup point and derive an energy estimate as a consequence.