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Exponential volumes of moduli spaces of hyperbolic surfaces

Alexander B. Goncharov, Zhe Sun

Abstract

A decorated surface S is an oriented topological surface with marked points on the boundary considered modulo the isotopy. We consider the moduli space of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. This space carries a volume form. Let us fix the set K of distances between the horocycles at the adjacent cusps, and the set L of lengths of boundary circles without cusps. We get a subspace M(S; K,L) with the induced volume form Vol(K,L). However, if the cusps are present, the volume of the space M(S; K,L) is infinite. We introduce the exponential volume form exp(-W)Vol(K,L), where W is a positive function on the moduli space, given by the sum over cusps of the hyperbolic areas enclosed between the cusp and the horocycle at the cusp. We prove that the exponential volume, defined as the integral of the exponential volume form over the moduli space M(S; K,L), is always finite. We suggest that the moduli spaces M(S; K,L) with the exponential volume forms are the true analogs of the classical moduli spaces of Riemann surfaces, with the Weil-Petersson volume forms. In particular, they should be relevant to the open string theory. We support this by proving an unfolding formula for the integrals of measurable functions multiplied by the exponential volume form. It expresses them as finite sums of similar integrals over moduli spaces for simpler surfaces. They generalise Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces.

Exponential volumes of moduli spaces of hyperbolic surfaces

Abstract

A decorated surface S is an oriented topological surface with marked points on the boundary considered modulo the isotopy. We consider the moduli space of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. This space carries a volume form. Let us fix the set K of distances between the horocycles at the adjacent cusps, and the set L of lengths of boundary circles without cusps. We get a subspace M(S; K,L) with the induced volume form Vol(K,L). However, if the cusps are present, the volume of the space M(S; K,L) is infinite. We introduce the exponential volume form exp(-W)Vol(K,L), where W is a positive function on the moduli space, given by the sum over cusps of the hyperbolic areas enclosed between the cusp and the horocycle at the cusp. We prove that the exponential volume, defined as the integral of the exponential volume form over the moduli space M(S; K,L), is always finite. We suggest that the moduli spaces M(S; K,L) with the exponential volume forms are the true analogs of the classical moduli spaces of Riemann surfaces, with the Weil-Petersson volume forms. In particular, they should be relevant to the open string theory. We support this by proving an unfolding formula for the integrals of measurable functions multiplied by the exponential volume form. It expresses them as finite sums of similar integrals over moduli spaces for simpler surfaces. They generalise Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces.

Paper Structure

This paper contains 69 sections, 47 theorems, 296 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Exponential volumes of the moduli spaces of ideal hyperbolic surfaces
  3. Ideal hyperbolic surfaces.
  4. The exponential volumes.
  5. Mirzakhani's volume formula.
  6. The first main result.
  7. The Laplace transform of exponential volumes and ${\cal B}-$functions
  8. The Laplace transform of exponential volumes.
  9. ${\cal B}-$function of a decorated surface ${\Bbb S}$.
  10. The function ${\cal B}_{\Bbb S}$ via ${\rm A}-$model.
  11. Neck recursion formula
  12. Unfolding exponential volumes
  13. Reduction to the case when $p$ is single cusp on a crown.
  14. Trouser legs and ideal triangles at a cusp $p$.
  15. Unfolding for a crown with a single cusp $p$.
  16. ...and 54 more sections

Key Result

Theorem 1.5

For any decorated surface ${\Bbb S}$, the exponential volume ${\rm Vol}_{{\mathcal{E}}}\mathcal{M}_{\Bbb S}({{\rm K}}, {\rm L})$ is finite.

Figures (35)

  • Figure 1: A decorated surface with a puncture and four marked points.
  • Figure 2: An ideal hyperbolic structure on a punctured disc with $4$ marked points is a hyperbolic crown with $4$ cusps and the neck geodesic $\gamma$, and a choice of a horocycle at each cusp, omitted on the picture.
  • Figure 3: An ideal hyperbolic surface. The horocycles around the cusps are shown in red.
  • Figure 5: A trouser leg ${\rm T}$ is a decorated surface given by an annulus with one speical boundary point. A geodesic trouser leg ${\rm T}$ carries an ideal hyperbolic structure with a geodesic loop $\ell_{\rm T}$ and a cusp with a horocycle. The length of $\ell_{\rm T}$ is $l$. The horocycle length of the bi-infinite geodesic is $\log {\rm K}^{-\frac{1}{2}}$. An ideal geodesic triangle $\tau$ with horocycles has the sides of horocycle lengths$(\log {\rm K}^{-\frac{1}{2}}_a, \log {\rm K}^{-\frac{1}{2}}_b, \log {\rm K}^{-\frac{1}{2}}_p)$.
  • Figure 6: The trouser leg ${\rm T}= {\rm T}({\rm K},l)$ at the cusp $p$ and the function $Q_{{\rm T}}({\rm K}, l)$.
  • ...and 30 more figures

Theorems & Definitions (97)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 1.6
  • proof
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 1.9
  • ...and 87 more