Singular Yamabe and Obata Problems

TL;DR

This paper explores a singular variant of the Yamabe problem in conformal geometry, analyzing solutions with sign-changing conformal factors and their geometric properties, extending classical problems like Obata's.

Contribution

It introduces a conformal Yamabe problem allowing singularities, characterizes zero loci of solutions, and connects them to Willmore energy and conformally invariant equations.

Findings

Zero locus is a smoothly embedded hypersurface.

In dimension three, zero locus minimizes Willmore energy.

Higher dimensions satisfy a conformally invariant Willmore analog.

Abstract

A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded separating hypersurface that, in dimension three, is necessarily a Willmore energy minimiser or, in higher dimensions, satisfies a conformally invariant analog of the Willmore equation. In any case the zero locus is critical for a conformal functional that generalises the total Q-curvature by including extrinsic data. These observations lead to some interesting global problems that include natural singular variants of a classical problem solved by Obata.