Classification of Steady Gradient Ricci-Yang-Mills Solitons on Surfaces

TL;DR

This paper constructs a continuous family of steady gradient Ricci-Yang-Mills solitons on surfaces, connecting known geometric structures and classifying all such solitons on surfaces.

Contribution

It introduces a one-parameter family of rotationally symmetric solitons on surfaces and proves their completeness and uniqueness.

Findings

The family connects the Hamilton cigar to the round sphere.

Complete solitons are characterized by this family.

Asymptotic behaviors vary with the parameter mbda.

Abstract

We construct string backgrounds in dimension 2 which connect the Hamilton cigar to the round sphere. Specifically, we construct a 1-parameter family of rotationally symmetric steady gradient Ricci-Yang-Mills solitons on surfaces, where we denote the parameter by $\lambda\in[-2,\infty)$. At $\lambda=-2$ is the Hamilton cigar, for $-2<\lambda<0$ the solitons are asymptotic to cylinders, at $\lambda=0$ is a complete noncompact soliton forming a cusp at infinity, and as $\lambda$ approaches infinity the family approaches a round point. Furthermore, we show any complete steady gradient Ricci-Yang-Mills soliton on a surface must come from this family.