Communications in Analysis and Geometry

Volume 31 (2023)

Number 3

Sharp entropy bounds for plane curves and dynamics of the curve shortening flow

Pages: 595 – 624

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n3.a3

Authors

Julius Baldauf (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass.,U.S.A.)

Ao Sun (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass.,U.S.A.; and Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, U.S.A.)

Abstract

We prove that a closed immersed plane curve with total curvature $2 \pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2 \pi m$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding–Minicozzi’s notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.

Full Text (PDF format)

Received 24 September 2018

Accepted 25 December 2020

Published 4 January 2024