New monotonicity formulas for the curve shortening flow in \({\mathbb {R}}^3\)

Abstract

For the curve shortening flow in \({\mathbb {R}}^3\) several new monotonicity formulas are derived. 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 planar curves.