Remarks on some isoperimetric properties of the k − 1 flow

Yu-Chu Lin and Dong-Ho Tsai

Vol. 260 (2012), No. 1, 197–213

DOI: 10.2140/pjm.2012.260.197

Abstract

We consider the evolution of a convex closed plane curve γ₀ along its inward normal direction with speed k − 1, where k is the curvature. This flow has the feature that it is the gradient flow of the (length − area) functional and has been previously studied by Chou and Zhu, and Yagisita. We revisit the flow and point out some interesting isoperimetric properties not discussed before.

We first prove that if the curve γ_t converges to the unit circle S¹ (without rescaling), its length L(t) and area A(t) must satisfy certain monotonicity properties and inequalities.

On the other hand, if the curve γ_t (assume γ₀ is not a circle) expands to infinity as t →∞ and we interpret Yagisita’s result in the right way, the isoperimetric difference L²(t) − 4πA(t) of γ_t will decrease to a positive constant as t →∞. Hence, without rescaling, the expanding curve γ_t will not become circular. It is asymptotically close to some expanding curve C_t, where C₀ is not a circle and each C_t is parallel to C₀. The asymptotic speed of C_t is given by the constant 1.

Keywords

curve shortening flow, isoperimetric inequality, plane curve evolution

Mathematical Subject Classification 2010

Primary: 51K05, 52A40

Milestones

Received: 29 November 2011

Revised: 24 June 2012

Accepted: 27 August 2012

Published: 11 October 2012