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Lorentz--Epstein surfaces and a Liouville action for positive curves

François Labourie, Jérémy Toulisse, Yilin Wang

Abstract

We investigate and define in this paper, in the context of the correspondence between anti-de Sitter $3$-space and $(1,1)$-conformal metrics, the analogs of $\cW$-volume, Epstein surfaces, and Liouville action. These notions were well-studied in the correspondence between $3d$-hyperbolic manifolds and $2d$ conformal metrics. We apply our construction to positive curves in flag manifolds equipped with a positive structure to obtain invariants of these curves that are finite in the case of piecewise circles.

Lorentz--Epstein surfaces and a Liouville action for positive curves

Abstract

We investigate and define in this paper, in the context of the correspondence between anti-de Sitter -space and -conformal metrics, the analogs of -volume, Epstein surfaces, and Liouville action. These notions were well-studied in the correspondence between -hyperbolic manifolds and conformal metrics. We apply our construction to positive curves in flag manifolds equipped with a positive structure to obtain invariants of these curves that are finite in the case of piecewise circles.
Paper Structure (50 sections, 42 theorems, 172 equations, 3 figures)

This paper contains 50 sections, 42 theorems, 172 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Conformal Lorentz surfaces
  3. Split structure
  4. Split vector spaces
  5. Split surfaces and Lorentz metric
  6. Compatible Lorentz structure
  7. Curvature of a Lorentz surface
  8. De Sitter surface
  9. Split annulus
  10. Isotropic surfaces
  11. The split structure of the Einstein torus
  12. Isotropic surfaces
  13. Dual isotropic surfaces and forms at infinity
  14. Isotropic, holonomic and Epstein surfaces
  15. Anti-de Sitter geometry
  16. ...and 35 more sections

Key Result

Theorem A

Let $(S,g)$ a Lorentzian surface, and $\phi$ a conformal immersion from $(S,g)$ to $\mathbf{Ein}^{1,1}$ satisfying a topological hypothesis. Then there exists a holonomic surface $\Sigma_g$ in the space of tangent vectors of norm $1$, $\mathsf U_+\mathbf{H}^{2,1}$, whose first fundamental form at in

Figures (3)

  • Figure 1: A polygonal curve $\mathcal{P}$ represented by $(\alpha_1,...,\alpha_6)$
  • Figure 2: Additivity of the crossratio area
  • Figure 3: From left to right: positive triple, quadruple, and curve

Theorems & Definitions (89)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4: Existence of isothermal coordinates
  • proof
  • ...and 79 more