Work

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

Year: 2026

Type: article

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.

Source: Discrete and Continuous Dynamical Systems

DOI: 10.3934/dcds.2026.105551

Authors: Jonah A. J. Duncan, Luc Nguyen

Institutions: University College London, University of Oxford

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: Engineering and Physical Sciences Research Council, Division of Mathematical Sciences

Award: Additional Funding for Mathematical Sciences: Heilbronn Institute for Mathematical Research