Non-uniqueness of curve shortening flow

TL;DR

This paper explores the uniqueness of curve shortening flow on symmetric surfaces and provides an example where non-uniqueness occurs under a specific non-flat metric, challenging existing assumptions.

Contribution

It formulates a conjecture on uniqueness for curve shortening flow and constructs a non-unique solution under a particular non-flat metric on the plane.

Findings

A non-flat metric on the plane can lead to non-uniqueness in curve shortening flow.

A properly embedded geodesic can evolve into multiple solutions under certain metrics.

The paper proposes a conjecture for uniqueness on symmetric surfaces.

Abstract

We formulate a uniqueness conjecture for curve shortening flow of proper curves on certain symmetric surfaces and give an example of a non-flat metric on the plane with respect to which curve shortening flow is not unique. That is, with respect to a suitably chosen metric, we construct a non-static solution to curve shortening flow starting from a properly embedded geodesic.