All two-dimensional expanding Ricci solitons

Abstract

The second author and H. Yin [Ars Inveniendi Analytica. DOI 10.15781/4x5c-9q97] have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a non-atomic Radon measure as a volume measure. This led to the discovery of a large array of new expanding Ricci solitons. In this paper, we use the recent uniqueness theory in this context, also developed by the second author and H. Yin [Proc. Lond. Math. Soc. 128:e12600 (2024)], to give a complete classification of all expanding Ricci solitons on surfaces. Along the way, we prove a converse to the existence theory that is not constrained to solitons: Every complete Ricci flow on a surface over a time interval $(0,\epsilon )$ admits a $t\downarrow 0$ limit within the class of admissible initial data. This makes surfaces the first non-trivial setting for Ricci flow in which a bijection can be given between the entire set of complete Ricci flows over maximal time intervals $(0,T)$, and a class of initial data that induce them.