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arXiv · 2602.23190

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

Jonah A. J. Duncan·Luc Nguyen
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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.

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Jonah A. J. Duncan, Luc Nguyen. 2026-02-26. Properties of hypersurface singular sets of solutions to the $\sigma_k$-Yamabe equation in the negative cone. https://arxiv.org/abs/2602.23190

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