Asymptotic behavior of complete conformal metric near singular boundary

TL;DR

This paper investigates the asymptotic behavior of solutions to the singular Yamabe problem near singular boundaries, providing optimal estimates and showing solutions approximate tangent cone solutions at boundary singularities.

Contribution

It offers new insights into the boundary behavior of solutions near singularities, extending understanding beyond smooth boundaries and including non-conformally flat backgrounds.

Findings

Solutions are well approximated by tangent cone solutions at singular boundary points.

Established optimal estimates for the background metric near singularities.

Extended analysis to non-conformally flat backgrounds.

Abstract

The boundary behavior of the singular Yamabe problem has been extensively studied near sufficiently smooth boundaries, while less is known about the asymptotic behavior of solutions near singular boundaries. In this paper, we study the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive the optimal estimates for the background metric which is not necessarily conformally flat. In particular, we prove that the solutions are well approximated by the solutions in tangent cones at singular points on the boundaries.