On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

TL;DR

This paper develops new methods from conformal geometry and scattering theory to construct solutions with singularities for the fractional Yamabe problem, extending classical gluing techniques to a non-local setting.

Contribution

It introduces a non-local version of the Mazzeo-Pacard gluing method for the fractional Yamabe problem, including analysis of a fractional Schrödinger equation with critical potential.

Findings

Constructed radial fast-decaying solutions using blow-up and bifurcation methods

Rewrote the non-local ODE via conformal geometry to suggest a non-local phase-plane analysis

Analyzed a fractional Schrödinger equation, establishing Green's function and Fredholm properties

Abstract

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schr\"{o}dinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.