Curve shortening flows on rotational surfaces generated by monotone convex functions

TL;DR

This paper investigates the behavior of curve shortening flows on negatively curved rotational surfaces in three-dimensional space, establishing conditions for the flow's long-term existence and the preservation of graph structure.

Contribution

It introduces new conditions involving Gauss curvature that ensure the curve remains a graph and proves long-time existence of the flow on such surfaces.

Findings

Curve remains a graph over parallels during flow

Comparison principle holds for the flow

Flow exists for long time under specified conditions

Abstract

In this paper, we study curve shortening flows on rotational surfaces in $\mathbb{R}^3$. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of embedded curve holds on them. Under these assumptions, we prove that the curve remains a graph over the parallels of the rotational surface along the flow. Also, we prove the comparison principle and the long-time existence of the flow.