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Classification of convex ancient free-boundary curve-shortening flows in the disc

Theodora Bourni and Mat Langford

Vol. 16 (2023), No. 9, 2225–2240
Abstract

Using a combination of direct geometric methods and an analysis of the linearization of the flow about the horizontal bisector, we prove that there exists a unique (modulo rotations about the origin) convex ancient curve-shortening flow in the disc with free boundary on the circle. This appears to be the first result of its kind in the free-boundary setting.

Keywords
curve-shortening flow, free-boundary, ancient solutions
Mathematical Subject Classification
Primary: 53E10
Milestones
Received: 10 November 2021
Revised: 23 February 2022
Accepted: 29 March 2022
Published: 11 November 2023
Authors
Theodora Bourni
Department of Mathematics
University of Tennessee
Knoxville, TN
United States
Mat Langford
Mathematical Sciences Institute
Australian National University
Canberra
Australia

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