Homogeneous Sobolev gradient flow of the length functional

TL;DR

This paper analyzes the gradient flow of the length functional on planar curves using homogeneous Sobolev metrics, establishing well-posedness, exponential decay, and preservation of geometric properties like immersion and convexity.

Contribution

It introduces a novel Sobolev gradient flow framework for length minimization, proving local well-posedness, global existence, and geometric property preservation.

Findings

Gradient flow is a reparametrisation-invariant nonlocal ODE.

Local Lipschitz continuity of the gradient ensures well-posedness.

Exponential decay of length and convergence to a constant map.

Abstract

We study the gradient flow of the length functional on the space of planar immersed closed curves, where the gradient is taken with respect to a family of homogeneous Sobolev $H^1$-type Riemannian metrics depending on parameters $\lambda>0$ and $a\in\mathbb{R}$. The gradient can be written explicitly in terms of arc-length convolution with the periodic Green's function for the second-order operator associated with the $H^1$ metric, and then the gradient flow is a reparametrisation-invariant nonlocal ODE. Working in the optimal low-regularity setting $W^{1,1}(\mathbb{S},\mathbb{R}^2)$, we show that the gradient is locally Lipschitz to obtain local well-posedness via the Picard--Lindel\"of theorem in Banach spaces. A time-reparametrisation reduces the analysis for general $a$ to the model case ${a=2}$, for which we obtain exponential decay of the length and global existence with uniform convergence in $W^{1,1}$ to a constant map. For $C^1$ immersed initial data we show that immersion is preserved for all time, and we further prove that if the initial curve bounds a convex set then convexity is also preserved by the flow.