The area preserving curve shortening flow with Neumann free boundary conditions

Abstract

We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.

Singularities of the area preserving curve shortening flow with a free boundary condition

“…We study the geometry of the limiting arc and get a contradiction to the assumptions. We refine results from [15] to show that the singularity is of type II. If the initial curve is convex we showed in [15] that the "Hamilton blowup" yields a grim reaper or half a grim reaper at a straight line.…”

“…The structure of this paper is as follows. In Section 2 we recall some results from [14,15] that we use in the proof of Theorem 1.1. We explain again how strongly the condition L 0 < d Σ influences the the behavior of γ(·,t) κds along the flow.…”

“…A bound on |κ| independent of T max is a consequence. If T max = ∞, then the bound on |κ| together with [15] imply subconvergence to a part of a circle that is possibly (partly) multicovered. We study the geometry of the limiting arc and get a contradiction to the assumptions.…”

“…The technique how to transform the free boundary problem into a standard Neumann boundary problem can be found in [17,18]. For our specific situation a sketch of the existence and regularity result is in [15,Proposition 2.4].…”

“…We recall some basic properties proved in [15]. Lemma 2.6 (Lemma 2.6, Lemma 2.11 and Corollary 2.14 [15]). Let γ 0 : [a, b] → R 2 be a smooth initial curve.…”