Existence and uniqueness to a fully non-linear version of the Loewner-Nirenberg problem

TL;DR

This paper establishes existence and uniqueness of a conformally flat metric with prescribed Schouten curvature on Euclidean domains, generalizing classical scalar curvature problems to fully non-linear equations with boundary and singularity conditions.

Contribution

It extends the Loewner-Nirenberg problem to fully non-linear curvature equations, providing existence, uniqueness, and boundary behavior results for conformally flat metrics.

Findings

Proves existence and uniqueness of solutions on smooth bounded domains.

Provides a sharp condition for metric divergence near higher codimension boundary parts.

Generalizes classical scalar curvature results to fully non-linear curvature equations.

Abstract

We consider the problem of finding on a given Euclidean domain $\Omega$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(\lambda(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partial\Omega$ is a smooth bounded hypersurface (of codimension one). When $\partial\Omega$ contains a compact smooth submanifold $\Sigma$ of higher codimension with $\partial\Omega\setminus\Sigma$ being compact, we also give a `sharp' condition for the divergence to infinity of the conformal factor near $\Sigma$ in terms of the codimension.