Curve shortening flow on Riemann surfaces with possible ambient conic singularities

TL;DR

This paper extends the understanding of curve shortening flow on Riemann surfaces, including those with conic singularities, by generalizing comparison techniques and analyzing the behavior near singularities.

Contribution

It introduces a generalized comparison function for Riemann surfaces with conic singularities and proves new results about the behavior of CSF near such singularities.

Findings

CSF cannot touch conic singularities with cone angles ≤ π

Reproves Gage-Hamilton-Grayson theorem on surfaces

Generalizes Huisken's comparison function to singular surfaces

Abstract

In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles $\leq \pi$.