A speed preserving Hilbert gradient flow for generalized integral Menger curvature

TL;DR

This paper introduces a novel Hilbert gradient flow for generalized integral Menger curvature that preserves curve speed and knot class, ensuring long-term existence, and demonstrates efficient numerical simulation methods.

Contribution

It establishes long-time existence and speed preservation for the flow, and develops a more efficient numerical simulation approach compared to existing methods.

Findings

Proves long-time existence of the flow.

Maintains knot class and curve speed during evolution.

Provides a more efficient numerical simulation technique.

Abstract

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case, and provide $C^{1,1}$-bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical $L^2$ gradient descent or other optimization methods.