Trajectory Surfaces of Framed Curvature Flow

TL;DR

This paper introduces the framed curvature flow, a new geometric flow that generalizes existing flows and analyzes the resulting surfaces with constant curvature properties.

Contribution

It defines the framed curvature flow, establishes existence and estimates, and studies the resulting trajectory surfaces with constant mean or Gaussian curvature.

Findings

Established local existence and global estimates for the flow.

Analyzed surfaces of constant mean curvature generated by the flow.

Analyzed surfaces of constant Gaussian curvature generated by the flow.

Abstract

This work introduces the framed curvature flow, a generalization of both the curve shortening flow and the vortex filament equation. Here, the magnitude of the velocity vector is still determined by the curvature, but its direction is given by an associated time-dependent moving frame. After establishing local existence and global estimates, we analyze the trajectory surfaces generated by different variations of this flow, specifically those leading to surfaces of constant mean or Gaussian curvature.