Complete $(2+1)$-dimensional Ricci flow spacetimes

Abstract

For a complete $(2+1)$-dimensional Ricci flow spacetime, we show that every worldline persists until the terminal time of the flow, and that at each time, the collection of worldlines not yet present is scattered. Consequently, only countably many worldlines can have inception time strictly larger than the initial time of the flow. Under non-negative curvature, every spatial slice must be diffeomorphic to a fixed surface and the spacetime is isomorphic to a classical Ricci flow on this surface. As an application, for any two-dimensional Ricci soliton, the completeness of the metric forces completeness of its soliton vector field.

arXiv comment26 pages. Substantially revised and expanded version, with strengthened results, new applications, and improved exposition throughout. Comments welcome