Uniqueness and nonuniqueness for Ricci flow on surfaces: Reverse cusp singularities

TL;DR

This paper investigates the uniqueness and nonuniqueness of Ricci flow evolutions on surfaces with cusps, demonstrating conditions under which the flow is uniquely determined or not, and constructing explicit examples of nonuniqueness.

Contribution

It extends the concept of initial conditions for Ricci flow on surfaces and establishes criteria for uniqueness, including explicit examples of nonuniqueness involving cusp geometries.

Findings

Surfaces with cusps can evolve with or without contracting the cusps.

Adding a noncollapsedness condition ensures uniqueness of the Ricci flow.

Explicit examples demonstrate nonuniqueness in Ricci flow evolution.

Abstract

We extend the notion of what it means for a complete Ricci flow to have a given initial metric, and consider the resulting well-posedness issues that arise in the 2D case. On one hand we construct examples of nonuniqueness by showing that surfaces with cusps can evolve either by keeping the cusps or by contracting them. On the other hand, by adding a noncollapsedness assumption for the initial metric, we establish a uniqueness result.