Curve shortening flow on Riemann surfaces with conical singularities

TL;DR

This paper investigates the behavior of the curve shortening flow on Riemann surfaces with conical singularities, establishing fundamental existence, uniqueness, and regularity results, along with convergence properties.

Contribution

It introduces a framework for analyzing the flow through singular points, proving short-term existence and stability of solutions on such surfaces.

Findings

Evolved curves remain fixed at singular points.

Established short-time existence and uniqueness.

Proved convergence and collapsing behavior.

Abstract

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some collapsing and convergence results.