A delayed interior area-to-height estimate for the Curve Shortening Flow

TL;DR

This paper extends the delayed interior regularity estimates for the Curve Shortening Flow, demonstrating interior graphical estimates and applications like initiating flows from weak initial data.

Contribution

It generalizes previous results to show interior graphical estimates and applies these to start flows from Radon measures without point masses.

Findings

Established interior graphical estimates for Curve Shortening Flow.

Proved existence of flows starting from weak initial data like Radon measures.

Extended delayed regularity framework to broader graphical settings.

Abstract

The principle of delayed parabolic regularity for the Curve Shortening Flow - that if two evolving curves bound a region of area $\mathcal A$, then, starting from time ${\mathcal A}/\pi$, the regularity of one curve is controllable in terms of the time elapsed, the area $\mathcal A$ and the regularity of the other curve - was proposed by Topping & the author in (Sobnack & Topping, 2024), where they also provided a number of graphical situations in which their delayed regularity framework is valid. In this paper, we generalise some of the results in (Sobnack & Topping, 2024) within the graphical setting, ultimately by showing that there holds an interior graphical estimate for the Curve Shortening Flow in the spirit of the proposed framework. We also provide a few applications of our estimate, such as the existence of Graphical Curve Shortening Flows starting weakly from Radon measures without point masses.