Classification of fractional, singular Yamabe metrics on a twice punctured sphere I

TL;DR

This paper classifies certain conformally flat, constant fractional Q-curvature metrics on a twice-punctured sphere, showing they are Delaunay metrics when the fractional order is close to 1, and establishes a sharp bound for the conformal factor.

Contribution

It proves that near s=1, all complete, conformally flat constant Q-curvature metrics on a twice-punctured sphere are Delaunay metrics, and provides a sharp a priori bound for their conformal factors.

Findings

Classification of metrics as Delaunay when s is close to 1

Establishment of a sharp a priori bound for the conformal factor

Identification of the family of Delaunay metrics as the only solutions in the specified setting

Abstract

The Delaunay metrics form a family of conformally flat, constant fractional Q-curvature metrics on a twice-punctured sphere. They are all (after a M\"obius transformation) rotationally symmetric and periodic, and admit several elegant variational descriptions. We prove that, when s is close to but less than 1, any complete, conformally flat constant Q-curvature metric on a twice-punctured sphere is a Delaunay metric. Along the way, we prove a sharp a priori bound for the conformal factor of these metrics, which may be of independent interest.