Delayed parabolic regularity for curve shortening flow

TL;DR

This paper investigates the regularity properties of curves evolving under curve shortening flow, establishing a principle relating the regularity of boundary curves and constructing solutions from minimal initial data.

Contribution

It introduces a new principle linking the regularity of boundary curves during flow and proves several related results, including the optimal timing for estimates.

Findings

Regularity of one boundary can be controlled by the other after a specific time

No estimates are valid before the time A/π

Constructs solutions from initial data that is only an L^1 function

Abstract

Given two curves bounding a region of area $A$ that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time $A/\pi$. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an $L^1$ function.