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Projective structures on curves and conformal geometry

Florin Belgun, Andrei Moroianu

TL;DR

This work establishes a precise equivalence between projective structures on curves and Laplace structures, via developing maps and holonomy in the $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ framework, and then capitalizes on this to classify open and closed projective curves through invariants such as winding numbers and holonomy generators. By tying these geometric structures to Hill's equation $x''+F x=0$ (with $F$ periodic) and Schwarzian calculus, the authors provide a detailed, invariant-based panorama of how curves sit inside ambient conformal/Möbius geometries. They prove that the Yamabe problem for curves in a conformal/Möbius ambient space has no general solution, since non-homogeneous Laplace structures cannot be tuned to have a constant zero-order term, despite the fact that every Laplace structure can be realized as ambient-induced. The results correct longstanding inaccuracies in the literature and illuminate the interplay between differential operators, projective holonomy, and ambient conformal geometry with concrete invariants such as holonomy classes, winding numbers, and resonance points.

Abstract

Projective structures on curves appear naturally in many areas of mathematics, from extrinsic conformal geometry to analysis, where the main problem is to find qualitative information about the solutions of Hill equations. In this paper, we describe in detail the correspondence between different equivalent definitions of projective structures and their isomorphism classes, correcting long-standing inexactitudes in the literature. As an application, we show that the {\em Yamabe problem for curves} in a conformal/Möbius ambient space has no solutions in general.

Projective structures on curves and conformal geometry

TL;DR

This work establishes a precise equivalence between projective structures on curves and Laplace structures, via developing maps and holonomy in the framework, and then capitalizes on this to classify open and closed projective curves through invariants such as winding numbers and holonomy generators. By tying these geometric structures to Hill's equation (with periodic) and Schwarzian calculus, the authors provide a detailed, invariant-based panorama of how curves sit inside ambient conformal/Möbius geometries. They prove that the Yamabe problem for curves in a conformal/Möbius ambient space has no general solution, since non-homogeneous Laplace structures cannot be tuned to have a constant zero-order term, despite the fact that every Laplace structure can be realized as ambient-induced. The results correct longstanding inaccuracies in the literature and illuminate the interplay between differential operators, projective holonomy, and ambient conformal geometry with concrete invariants such as holonomy classes, winding numbers, and resonance points.

Abstract

Projective structures on curves appear naturally in many areas of mathematics, from extrinsic conformal geometry to analysis, where the main problem is to find qualitative information about the solutions of Hill equations. In this paper, we describe in detail the correspondence between different equivalent definitions of projective structures and their isomorphism classes, correcting long-standing inexactitudes in the literature. As an application, we show that the {\em Yamabe problem for curves} in a conformal/Möbius ambient space has no solutions in general.

Paper Structure

This paper contains 15 sections, 24 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Curves in conformal geometry
  3. Laplace structures on curves, projective parametrizations and Hill's equation
  4. Laplace structures on curves
  5. Laplace structures on curves in conformal/Möbius geometry
  6. Projective structures on curves
  7. The developing map and holonomy of a closed curve
  8. The action of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ on $\widetilde{\mathbb{R}\mathbb{P}^1}$
  9. Open projective curves
  10. The isomorphism classes of closed projective curves
  11. The conjugacy classes in $\widetilde{\mathrm{SL}}(2,\mathbb{R})$
  12. Classification of projective structures on closed curves (with or without a fixed orientation)
  13. Classification of Laplace structures on closed curves
  14. Geometric properties of closed projective curves
  15. The Yamabe problem for curves in conformal geometry

Key Result

Lemma 3.3

The difference between the Hessians on $L^k$ of two arbitrary connections $\nabla,\nabla'$ on a curve $C$ is a zero-order operator if and only if $k=1/2$.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 51 more