Properties of hypersurface singular sets of solutions to the $\sigma_k$-Yamabe equation in the negative cone

TL;DR

This paper studies the geometric and analytical properties of solutions to the $\sigma_k$-Yamabe equation with hypersurface singularities, revealing PDE relations and minimality conditions in the negative cone setting.

Contribution

It establishes PDE conditions for solutions with hypersurface singularities and demonstrates minimality for the case $k=2$, advancing understanding of singular solutions in conformal geometry.

Findings

Trace and normal derivatives satisfy a PDE along the hypersurface

Hypersurface is minimal for $k=2$ solutions

Addresses formal expansion near hypersurfaces

Abstract

We consider conformally flat Lipschitz viscosity solutions to the $\sigma_k$-Yamabe equation in the negative cone which admit smooth hypersurface singularities. Under natural regularity assumptions (that are satisfied by solutions to the $\sigma_k$-Loewner-Nirenberg problem on annuli, for example), we first prove that the trace and normal derivatives of such a solution along the hypersurface satisfy a certain PDE. For $k=2$, we also show that the hypersurface is minimal with respect to the Lipschitz solution and address some questions related to the formal expansion of the solution near the hypersurface.