Regularity of viscosity solutions of the $\sigma_k$-Yamabe-type Problem for $k>n/2$

TL;DR

This paper investigates the regularity of viscosity solutions to the $\sigma_k$-Yamabe problem for $k>n/2$, establishing smoothness outside negligible sets and existence results under certain conditions.

Contribution

It proves regularity of solutions in specific cases and demonstrates existence of Lipschitz solutions with smoothness outside measure-zero sets for the general case.

Findings

Lipschitz viscosity solutions are smooth outside measure-zero sets for $k=n$ or conformally flat manifolds with $k>n/2$.

Existence of Lipschitz viscosity solutions that are smooth outside measure-zero sets for general $k>n/2$ under certain assumptions.

The regularity results extend understanding of the $\sigma_k$-Yamabe problem in the negative cone case.

Abstract

We study the regularity of Lipschitz viscosity solutions to the $\sigma_k$ Yamabe problem in the negative cone case. If either $k=n$ or the manifold is conformally flat and $k>n/2$, we prove that all Lipschitz viscosity solutions are smooth away from a closed set of measure zero. For the general $k>n/2$ case, under certain assumptions, we prove the existence of a Lipschitz viscosity solution that is smooth away from a closed set of measure zero.