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

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.