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The curve-lengthening flow in inversive geometry

Ben Andrews, Glen Wheeler

TL;DR

The paper introduces an invariant gradient flow for the length functional of co-compact curves in inversive geometry, proving global existence and exponential convergence to a loxodromic limit. It builds a comprehensive inversive differential-geometry framework (invariants, Gauss map into $PSL(2,\mathbb{C})$, invariant parameter, Serret–Frenet structure) and derives a sixth-order, invariant curve-lengthening flow $\partial_t X=-Q_s\mathcal{N}$ whose dynamics are governed by a high-order PDE for the fundamental invariant $Q$. Key contributions include robust a priori estimates (energy decay in $L^2$, higher-derivative bounds), proof of long-time existence, and exponential convergence to $L$-cocompact loxodromic curves; the results extend to non-smooth initial data via a smoothing-approximation scheme. The work provides a Möbius-invariant analogue of curvature-driven curve flows, yielding precise asymptotic behavior and potential applications in conformal and Möbius-invariant geometric analysis.

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).

The curve-lengthening flow in inversive geometry

TL;DR

The paper introduces an invariant gradient flow for the length functional of co-compact curves in inversive geometry, proving global existence and exponential convergence to a loxodromic limit. It builds a comprehensive inversive differential-geometry framework (invariants, Gauss map into , invariant parameter, Serret–Frenet structure) and derives a sixth-order, invariant curve-lengthening flow whose dynamics are governed by a high-order PDE for the fundamental invariant . Key contributions include robust a priori estimates (energy decay in , higher-derivative bounds), proof of long-time existence, and exponential convergence to -cocompact loxodromic curves; the results extend to non-smooth initial data via a smoothing-approximation scheme. The work provides a Möbius-invariant analogue of curvature-driven curve flows, yielding precise asymptotic behavior and potential applications in conformal and Möbius-invariant geometric analysis.

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).

Paper Structure

This paper contains 16 sections, 16 theorems, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Inversive Differential geometry of curves
  3. Inversive transformations
  4. The Inversive Gauss map
  5. Invariant parameter, tangent and normal vectors
  6. Serret-Frenet equations
  7. Expansion of the Gauss map in invariant parameter
  8. Variation equations
  9. Co-compactness
  10. The evolution equations
  11. The Inversive curve-lengthening flow
  12. L2 bound on the invariant curvature
  13. Bounds on higher derivatives
  14. Long time existence
  15. Exponential convergence
  16. ...and 1 more sections

Key Result

Theorem 1

Let $X_0:\ \mathbb{R}\to\mathbb{C}\mathbb{P}^1$ be an admissible $L$-cocompact curve for which $\int Q^2\,ds<\infty$. Then there exists a unique solution $X:\ \mathbb{R}\times[0,\infty)\to\mathbb{C}\mathbb{P}^1$ of the inversive curve lengthening flow eq:ICLF which is smooth for $t>0$ and in $C^{0,1

Figures (3)

  • Figure 1: Three loxodromes on $\mathbb{CP}^1$. They are determined by the maps $\left[w_0(u)w_1(u)\right] = \left[\exp((a+i)u)1\right]$ in homogeneous coordinates on $\mathbb{CP}^1$.
  • Figure 2: Stereographic projection of the curves from $\mathbb{CP}^1$ in Figure \ref{['figLox1']}. In the complex plane their parametrisation is $u\mapsto \exp((a + i)u)$.
  • Figure 3: An illustration of three loxodromic curves with $n=0, 1$ and $2$. The integer $n$ counts the winding in each homotopy class.

Theorems & Definitions (35)

  • Theorem 1
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Definition 1
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • ...and 25 more