On the Negative Case of the Singular Yamabe Problem

TL;DR

This paper investigates the conditions under which complete metrics with constant negative scalar curvature exist in the negative case of the Singular Yamabe Problem, emphasizing the role of the tangent structure of the excluded set.

Contribution

It demonstrates that the existence of such metrics depends on the tangent cone structure of the closed set, extending previous dimension-based criteria.

Findings

Existence depends on tangent cone structure.

Metrics do not exist if tangent cone dimension is less than (n-2)/2.

Results generalize known dimension conditions for smooth submanifolds.

Abstract

The negative case of the Singular Yamabe Problem concerns the existence and behavior of complete metrics with constant negative scalar curvature on the complement of a closed set in a compact Riemannian manifold which are conformally equivalent to a smooth metric on this compact manifold. When the closed set is a smooth submanifold, it is known by the results of Loewner-Nirenberg and Aviles-McOwen that there exists such a complete metric if and only if $d > (n-2)/2$, and in general the Hausdorff dimension of the set must be at least $(n-2)/2$. In this paper, we show that the existence of such a complete conformal metric with constant negative scalar curvature depends on the tangent structure of the closed set. Specifically, provided the set has a nice tangent cone at a point, we show that when the dimension of this tangent cone is less than $(n-2)/2$ there can not exist such a negative Singular Yamabe metric.