Communications in Analysis and Geometry

Volume 28 (2020)

Number 8

The Second of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

Evolution of locally convex closed curves in the area-preserving and length-preserving curvature flows

Pages: 1863 – 1894

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n8.a5

Authors

Natasa Sesum (Department of Mathematics, Rutgers University, Pitscataway, New Jersey, U.S.A.)

Dong-Ho Tsai (Department of Mathematics, National Tsing Hua University, Hsinchu,Taiwan)

Xiao-Liu Wang (School of Mathematics, Southeast University, Nanjing, China)

Abstract

We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an $m$-fold circle as time goes to infinity. For the area-preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in finite time or not, while for the length-preserving flow, it is the positivity of an energy associated with initial curve that plays such a role.

Full Text (PDF format)

The first author thanks the NSF support in DMS-1056387. The second author is supported by NCTS and MoST of Taiwan with grant number 108-2115-M-007-013-MY2. The third author is supported by the Fundamental Research Funds for the Central Universities 2242015R30012, the NSF of China 11101078,11871148 and the Natural Science Foundation of Jiangsu Province BK20161412.

Received 12 July 2017

Accepted 13 April 2020

Published 8 January 2021