The level-set flow of the topologist's sine curve is smooth

TL;DR

This paper demonstrates that the level-set flow of the topologist's sine curve results in a smooth closed curve, providing the first example of a non-locally-connected boundary evolving instantaneously into smoothness.

Contribution

It introduces the first example of a non-locally-connected set whose level-set flow becomes smooth instantly, expanding understanding of flow behavior for complex boundaries.

Findings

Level-set flow of the topologist's sine curve is smooth.

Non-locally-connected boundaries can evolve into smooth curves.

Instantaneous smoothing occurs for certain non path-connected sets.

Abstract

In this note we prove that the level-set flow of the topologist's sine curve is a smooth closed curve. In previous work it was shown by the second author that under level-set flow, a locally-connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we give the first example of a domain whose boundary is not locally-connected for which the level-set flow is instantaneously smooth. Our methods also produce an example of a non path-connected set that instantly evolves into a smooth closed curve.