Ricci flow on Finsler surfaces

TL;DR

This paper establishes the short-term existence and uniqueness of Ricci flow solutions on Finsler surfaces by analyzing the Ricci-DeTurck flow and employing diffeomorphisms, supported by illustrative examples.

Contribution

It introduces the first analysis of Ricci flow on Finsler surfaces, proving short-time existence and uniqueness through the Ricci-DeTurck approach.

Findings

Proved short-time existence of Ricci flow on Finsler surfaces.

Established uniqueness of solutions for the Ricci flow.

Provided examples illustrating the flow on specific Finsler surfaces.

Abstract

Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to the original Ricci flow by pulling back the solution of the Ricci-DeTurck flow using appropriate diffeomorphisms. In the end, we illustrate this argument with some examples.