A singular Yamabe problem on manifolds with solid cones

TL;DR

This paper investigates the existence of conformal metrics with negative scalar and mean curvature on non-compact manifolds with boundary, constructed by removing submanifolds from spaces modeled on solid cones, extending classical Yamabe problem results.

Contribution

It establishes a precise condition for the existence of such metrics on manifolds with singularities, generalizing the singular Yamabe problem to new geometric settings.

Findings

Existence of conformal metrics if and only if d > (n-2)/2

Construction of metrics on manifolds with boundary modeled on solid cones

Extension of classical Yamabe problem results to non-compact, singular settings

Abstract

We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d>(n-2)/2. Our main theorem is inspired by the classical results by Aviles-McOwen and Loewner-Nirenberg known in the literature as the "singular Yamabe problem".