Stabilization Technique Applied To Curve Shortening Flow in the Plane

Abstract

The method proposed by T. I. Zelenjak is applied to the mean curvature flow in the plane. A new type of monotonicity formula for star-shaped curves is obtained.Date: 1/DEC/2014.

Stabilization technique applied to curve shortening flow in $\mathbb{R}^3$

“…Remark 1. It has been shown in [12] that using the stabilization technique one can not only verify but actually also re-discover Huisken's monotonicity formula in R 2 . This is true also in 3D as one can see in the next sections.…”

“…In the case of the Huisken's formula functions F and ρ depend only on absolute values of ξ and η. Generalizing the approach developed in [12] for plane curves we are looking for formulae which depend on the angle between ξ and η.…”

“…Taking into account the matrices (12) we look for the functions F and ρ of the form F (ξ, η) = a(|ξ|)|η|f (ψ) and ρ(ξ, η) = a(|ξ|)|η|g(ψ), with ψ = arcsin ξ, η |ξ||η| being the angle between the position vector ξ = γ and the plane orthogonal to the tangent η = γ . The functions f (ψ), g(ψ) and a(r) will be specified in upcoming sections.…”

“…Huisken's monotonicity formula plays a crucial role in the theory of flows driven by mean curvature, in particular curve shortening flow. In this article we are going to develop ideas from [12] for the curve shortening flow in the plain to obtain a method for deriving monotonicity formulae in 3D. This idea is based on an article by T. Zelenyak [13], where the author derived general monotonicity formulae for 1D parabolic problems on an interval.…”

A short proof of monotonicity formula for curve shortening flow in 3D

“…This formula is the 3D codimension-2 analogue of the famous Huisken's formula (see [1]). The proof is based on ideas introduced by Zelenjak (see [3]) for parabolic boundary value problems and adapted by the author in 2D curve shortening context in [2].…”