Moduli of spherical tori with one conical point

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 $2\pi \vartheta$. 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 $m>1$.

arXiv comment64 pages, 9 figures

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Cited by in corpus (2)

• Metrics of constant positive curvature with conic singularities. A survey
• The space of Schwarz-Klein triangles