All two-dimensional expanding Ricci solitons

TL;DR

This paper classifies all expanding Ricci solitons on surfaces using recent uniqueness results, and establishes a correspondence between Ricci flows and initial data in this setting.

Contribution

It provides a complete classification of all expanding Ricci solitons on surfaces and proves a converse existence result linking Ricci flows to initial data.

Findings

Complete classification of expanding Ricci solitons on surfaces.

Every complete Ricci flow on a surface has a well-defined initial data limit.

Surfaces are the first setting with a bijection between Ricci flows and initial data.

Abstract

The second author and H. Yin have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a nonatomic 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, 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 $(0,\varepsilon)$ admits a $t\downarrow 0$ limit within the class of admissible initial data. This makes surfaces the first nontrivial 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 induces them.