Solutions to the singular $\sigma_2-$Yamabe problem with isolated singularities

TL;DR

This paper constructs solutions to the singular -Yamabe problem with isolated singularities on certain manifolds, using advanced perturbation and gluing techniques to handle high-order Weyl tensor vanishing.

Contribution

It provides the first existence results for the singular -Yamabe problem with isolated singularities on manifolds with specific curvature conditions.

Findings

Existence of solutions with prescribed singularities

Solutions constructed via perturbation and gluing methods

Applicable to manifolds with high-order Weyl tensor vanishing

Abstract

Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular $\sigma_k-$Yamabe problem asks for a complete metric $g$ on $M\backslash X$ conformal to $g_0$ with constant $\sigma_k-$curvature. The $\sigma_k-$curvature is defined as the $k-$th elementary symmetric function of the eigenvalues of the Schouten tensor of a Riemannian metric. The main goal of this paper is to find solutions to the singular $\sigma_2-$Yamabe problem with isolated singularities in any compact non-degenerate manifold such that the Weyl tensor vanishing to sufficiently high order at the singular point. We will use perturbation techniques and gluing methods.