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Moduli of spherical tori with one conical point

Alexandre Eremenko, Gabriele Mondello, Dmitri Panov

Abstract

In this paper we determine the topology of the moduli space $\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannian metric of constant curvature $1$ and one conical point of angle $2π\vartheta$. In particular, for $\vartheta\in (2m-1,2m+1)$ non-odd, $\mathcal{MS}_{1,1}(\vartheta)$ is connected, has orbifold Euler characteristic $-m^2/12$, and its topology depends on the integer $m>0$ only. For $\vartheta=2m+1$ odd, $\mathcal{MS}_{1,1}(2m+1)$ has $\lceil{m(m+1)/6}\rceil$ connected components. For $\vartheta=2m$ even, $\mathcal{MS}_{1,1}(2m)$ has a natural complex structure and it is biholomorphic to $\mathbb{H}^2/G_m$ for a certain subgroup $G_m$ of $\mathrm{SL}(2,\mathbb{Z})$ of index $m^2$, which is non-normal for $m>1$.

Moduli of spherical tori with one conical point

Abstract

In this paper we determine the topology of the moduli space of surfaces of genus one with a Riemannian metric of constant curvature and one conical point of angle . In particular, for non-odd, is connected, has orbifold Euler characteristic , and its topology depends on the integer only. For odd, has connected components. For even, has a natural complex structure and it is biholomorphic to for a certain subgroup of of index , which is non-normal for .

Paper Structure

This paper contains 41 sections, 73 theorems, 17 equations, 8 figures.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. $\vartheta$ not an odd integer
  4. $\vartheta$ odd integer
  5. $\vartheta$ even integral
  6. Analytic representation of spherical metrics
  7. The idea of the proof of Theorem \ref{['mainonodd']}
  8. Content of the paper
  9. Acknowledgement
  10. Voronoi diagram and proof of Theorem \ref{['twotrianglesTH']}
  11. Properties of Voronoi graphs, functions and domains
  12. The circumcenters of balanced triangles
  13. Isometric conformal involutions on tori
  14. Proof of Theorem \ref{['twotrianglesTH']}
  15. Balanced spherical triangles
  16. ...and 26 more sections

Key Result

Theorem A

Take $\vartheta\in (1,\infty)$ that is not an odd integer and set $m=\lfloor{\frac{\vartheta+1}{2}}\rfloor$. The moduli space $\mathcal{MS}_{1,1}(\vartheta)$ of spherical tori with a conical point of angle $2\pi\vartheta$ is a connected orientable two-dimensional orbifold of finite type with the fol

Figures (8)

  • Figure 1: Voronoi graphs on a sphere with three conical points
  • Figure 2: Three types of spheres
  • Figure 3: Voronoi graphs of balanced triangles
  • Figure 4: Trefoil case
  • Figure 5: Eight graph case
  • ...and 3 more figures

Theorems & Definitions (182)

  • Theorem A: Topology of $\mathcal{MS}_{1,1}(\vartheta)$ for $\vartheta$ not odd
  • Remark 1.1: Orbifold structure and isometric involution
  • Definition 1.2: Spherical polygons
  • Definition 1.3: Balanced triangles
  • Theorem B: Canonical decomposition of a spherical torus for non-odd $\vartheta$
  • Theorem C: Topology of $\mathcal{MS}_{1,1}(2m+1)^\sigma$
  • Remark 1.5
  • Theorem D: Topology of $\mathcal{MS}_{1,1}(2m+1)$
  • Theorem E: Canonical decomposition of a spherical torus with odd $\vartheta$
  • Theorem F: $\mathcal{MS}_{1,1}(2m)$ is a Belyi curve
  • ...and 172 more