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

TL;DR

This paper extends the stabilization technique to curve shortening flow in three-dimensional space, deriving new monotonicity formulas that relate the curve's geometry to its energy and angular properties.

Contribution

It introduces novel monotonicity formulas for curve shortening flow in b2^3, generalizing classical results and incorporating stabilization methods.

Findings

Derived new monotonicity formulas involving angle-dependent energy terms.

Generalized Huisken's monotonicity formula to 3D curves.

Extended previous planar curve formulas to spatial curves.

Abstract

We apply the stabilization technique, developed by T. Zelenyak in 1960s for parabolic equations, on curve shortening flow in $\R^3$, and derive several new monotonicity formulas. All of them share one main feature: the dependence of the "energy" term on the angle between the position vector and the plane orthogonal to the tangent vector. The first formula deals with the projection of the curve on the unit sphere, and computes the derivative of its length. The second formula is the generalization of the classical formula of G. Huisken, while the third one is the generalization of the monotonicity formula with logarithmic terms previously derived by the author for plane curves.