Conformal metrics with prescribed fractional scalar curvature on conformal infinities with positive fractional Yamabe constants

TL;DR

This paper introduces and solves the prescribed fractional scalar curvature problem on conformal infinities of asymptotically hyperbolic manifolds, extending the fractional Yamabe problem to endpoint cases with new existence results.

Contribution

It formulates the prescribed fractional scalar curvature problem and provides existence solutions under various geometric conditions, including endpoint fractional Yamabe cases.

Findings

Existence of smooth solutions for prescribed fractional scalar curvature.

Solutions obtained for endpoint fractional Yamabe problem cases.

Extension of fractional Yamabe problem to non-umbilic conformal infinities.

Abstract

Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various geometric conditions on $X$ and $M$. We also obtain the existence results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$, $\gamma = 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to be smooth on $M$.