Smoothing a measure on a Riemann surface using Ricci flow

TL;DR

This paper develops a Ricci flow framework for Riemann surfaces with measure-based initial data, introduces new Ricci soliton examples, and explores the regularity of flows at initial time.

Contribution

It formulates existence results for Ricci flow with measure initial data, constructs new Ricci soliton examples, and addresses flow regularity at initial time.

Findings

Established existence of Ricci flow with Radon measure initial data.

Constructed the first nongradient Kaehler Ricci soliton.

Provided a counterexample to smoothness at initial time for certain flows.

Abstract

We formulate and solve the existence problem for Ricci flow on a Riemann surface with initial data given by a Radon measure as volume measure. The theory leads us to a large class of new examples of nongradient expanding Ricci solitons, including the first example of a nongradient Kaehler Ricci soliton. It also settles the question of whether a smooth flow for positive time that attains smooth initial data in a distance metric sense must be smooth down to the initial time. We disprove this by giving an example of a complete Ricci flow starting with the Euclidean plane that is not the static solution.