Admissibility, Topology, and the $\sigma_k$-Loewner--Nirenberg problem

Hao Fang; Matthew J. Gursky

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arXiv · 2606.16033

Admissibility, Topology, and the $\sigma_k$-Loewner--Nirenberg problem

Hao Fang·Matthew J. Gursky

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Abstract

The $\sigma_k$-Loewner--Nirenberg problem is a fully nonlinear generalization of the classical Loewner--Nirenberg problem of constructing complete conformal metrics with constant negative scalar curvature in the interior of domains. For the $\sigma_k$-version, once $k > 1$, one must impose a negative admissibility condition to ensure ellipticity of the equation. In this note, we exhibit topological obstructions to admissibility and illustrate this with examples.

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Hao Fang, Matthew J. Gursky. 2026-06-14. Admissibility, Topology, and the $\sigma_k$-Loewner--Nirenberg problem. https://arxiv.org/abs/2606.16033

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