Some existence results for the modified binormal curvature flow equation

TL;DR

This paper proves existence and uniqueness for a generalized binormal curvature flow where the speed depends on parametrization, time, and position, extending classical results through a Schrödinger map approach.

Contribution

It introduces a generalized form of the binormal curvature flow with variable speed, broadening the understanding of curve evolution in three-dimensional space.

Findings

Established existence and uniqueness for the generalized flow.

Connected the flow to a Schrödinger map formulation.

Extended classical binormal flow results to more general settings.

Abstract

We establish existence and uniqueness results for the modified binormal curvature flow equation that generalizes the binormal curvature flow equation for a curve in $\R^3.$ In this generalization, the velocity of the curve is still directed along the binormal vector, but the magnitude of the speed is allowed to depend on th parametrization of the curve, the time and the position of the point in the space. We achieve our objective via a generalized form of the Schr\"odinger map equation.