Properties of hypersurface singular sets of solutions to the $ \sigma_k $ -Yamabe equation in the negative cone
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.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Nonlinear Partial Differential Equations
Geometry and complex manifolds
Publication details
- Journal / source
- Discrete and Continuous Dynamical Systems
- Volume
- 61
- Issue
- 0
- Publication date
- 2 November 2026
- Language
- EN