The curve-lengthening flow in inversive geometry

TL;DR

This paper studies a special geometric flow in inversive geometry, proving long-term existence and convergence of curves to loxodromic shapes under certain conditions.

Contribution

Introduces an invariant gradient flow for the length functional in inversive geometry and proves its solutions exist globally and converge to specific curves.

Findings

Solutions exist for all time

Curves converge to loxodromic shapes

Flow preserves invariance properties

Abstract

We consider an invariant gradient flow for the invariant length functional for co-compact curves in inversive geometry, and prove that solutions exist for all time and converge to loxodromic curves, provided the initial curve is admissible (so that the invariant length element is well defined).