Grim Raindrop: A Translating Solution to Curve Diffusion Flow

TL;DR

This paper proves the existence of a translating solution to the higher-order curve diffusion flow in the plane, extending understanding of geometric flows beyond curve shortening flow.

Contribution

It introduces the first known translating solution to the curve diffusion flow, a higher-order geometric evolution equation.

Findings

Existence of a properly immersed translating solution established.

Extends the theory of geometric flows to higher-order cases.

Provides a foundation for future studies of curve diffusion dynamics.

Abstract

We show the existence of a properly immersed translating solution to curve diffusion flow in the plane. Curve diffusion flow is a higher order version of curve shortening flow, namely \[ \left( \frac{dX}{dt}\right) ^{\perp}=-\kappa_{ss}N. \]