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

Jonah A. J. Duncan, Luc Nguyen

Abstract

We consider conformally flat Lipschitz viscosity solutions to the $σ_k$-Yamabe equation in the negative cone which admit smooth hypersurface singularities. Under natural regularity assumptions (that are satisfied by solutions to the $σ_k$-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.

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

Abstract

We consider conformally flat Lipschitz viscosity solutions to the -Yamabe equation in the negative cone which admit smooth hypersurface singularities. Under natural regularity assumptions (that are satisfied by solutions to the -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 , 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.
Paper Structure (6 sections, 7 theorems, 88 equations)

This paper contains 6 sections, 7 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['A']} and \ref{['D']}
  3. A preliminary lemma on the ordering of $\nabla_\nu w^\pm$
  4. Proof of Theorem \ref{['A']}
  5. Proof of Theorem \ref{['D']}
  6. Proof of Proposition \ref{['B']}

Key Result

Theorem 1.2

Suppose that $w$ is a viscosity solution to 6 admitting a hypersurface $\Sigma$ of singular points in $B_\varepsilon$. Then for $\alpha \in \{\nabla_\nu w^\pm\}$,

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • ...and 9 more