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 admits a 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 , and a class of initial data that induce them.