Solutions of the singular Yamabe problem near singular boundaries

TL;DR

This paper studies the behavior of solutions to the singular Yamabe problem near singular boundaries, providing optimal estimates and approximations for solutions in non-conformally flat settings with conical structures.

Contribution

It extends existing results by analyzing solutions near singular boundaries without assuming conformal flatness, using tangent cone approximations.

Findings

Solutions are well approximated by tangent cone solutions.

Optimal estimates for solutions near singular boundaries.

Results apply to Lipschitz domains with conical structures.

Abstract

In this paper, we investigate the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive optimal estimates, where the background metrics are not assumed to be conformally flat. Specifically, we demonstrate that for a wide class of Lipschitz domains with asymptotic conical structure, the local positive solutions are well approximated by the positive solutions in the tangent cones at singular boundary points. This extends the results of [10, 12, 26].