Isolated singularities for a semilinear equation for the fractional Laplacian arising in conformal geometry

TL;DR

This paper investigates isolated singularities in a semilinear fractional Laplacian equation within conformal geometry, using geometric reformulation and phase portrait analysis for a non-local ODE.

Contribution

It introduces a geometric framework and phase portrait method to analyze singularities in fractional Laplacian equations relevant to conformal geometry.

Findings

Characterization of isolated singularities in the fractional Laplacian context

Development of a phase portrait approach for non-local ODEs

Insights into singular metrics with constant fractional curvature

Abstract

We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.