Moduli of spherical tori with one conical point

Eremenko, Alexandre¹ ; Mondello, Gabriele² ; Panov, Dmitri³

¹ Department of Mathematics, Purdue University, West Lafayette, IN, United States
² Dipartimento di Matematica Guido Castelnuovo, Sapienza Università of Roma, Roma, Italy
³ Department of Mathematics, King’s College London, London, United Kingdom

Geometry & topology, Tome 27 (2023) no. 9, pp. 3619-3698.

Résumé

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

Détail

DOI : 10.2140/gt.2023.27.3619

Keywords: spherical surfaces, moduli spaces, conical points, Belyi curves

Bibliographie

[1] D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) | DOI

[2] C L Chai, C S Lin, C L Wang, Mean field equations, hyperelliptic curves and modular forms, I, Camb. J. Math. 3 (2015) 127 | DOI

[3] C C Chen, C S Lin, Mean field equation of Liouville type with singular data : topological degree, Comm. Pure Appl. Math. 68 (2015) 887 | DOI

[4] Q Chen, W Wang, Y Wu, B Xu, Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces, Pacific J. Math. 273 (2015) 75 | DOI

[5] Z Chen, T J Kuo, C S Lin, Existence and non-existence of solutions of the mean field equations on flat tori, Proc. Amer. Math. Soc. 145 (2017) 3989 | DOI

[6] Z Chen, C S Lin, Critical points of the classical Eisenstein series of weight two, J. Differential Geom. 113 (2019) 189 | DOI

[7] D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, 5, Math. Soc. Japan (2000) | DOI

[8] B Delaunay, Sur la sphère vide, Bull. Acad. Sci. URSS (1934) 793

[9] A Eremenko, Metrics of positive curvature with conic singularities on the sphere, Proc. Amer. Math. Soc. 132 (2004) 3349 | DOI

[10] A Eremenko, Metrics of constant positive curvature with four conic singularities on the sphere, Proc. Amer. Math. Soc. 148 (2020) 3957 | DOI

[11] A Eremenko, A Gabrielov, On metrics of curvature 1 with four conic singularities on tori and on the sphere, Illinois J. Math. 59 (2015) 925

[12] A Eremenko, A Gabrielov, The space of Schwarz–Klein spherical triangles, J. Math. Phys. Anal. Geom. 16 (2020) 263 | DOI

[13] A Eremenko, A Gabrielov, G Mondello, D Panov, Moduli spaces for Lamé functions and abelian differentials of the second kind, Commun. Contemp. Math. 24 (2022) 2150028 | DOI

[14] A Eremenko, V Tarasov, Fuchsian equations with three non-apparent singularities, Symmetry Integrability Geom. Methods Appl. 14 (2018) 058 | DOI

[15] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (2007) | DOI

[16] J Harer, D Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986) 457 | DOI

[17] F Klein, Vorlesungen über die hypergeometrische Funktion, 39, Springer (1933) 344 | DOI

[18] L Li, J Song, B Xu, Irreducible cone spherical metrics and stable extensions of two line bundles, Adv. Math. 388 (2021) 107854 | DOI

[19] C S Lin, C L Wang, Elliptic functions, Green functions and the mean field equations on tori, Ann. of Math. 172 (2010) 911 | DOI

[20] C S Lin, C L Wang, Mean field equations, hyperelliptic curves and modular forms, II, J. Éc. polytech. Math. 4 (2017) 557 | DOI

[21] R S Maier, Lamé polynomials, hyperelliptic reductions and Lamé band structure, Philos. Trans. Roy. Soc. Lond. Ser. A 366 (2008) 1115 | DOI

[22] G Mondello, D Panov, Spherical metrics with conical singularities on a 2–sphere : angle constraints, Int. Math. Res. Not. 2016 (2016) 4937 | DOI

[23] G Mondello, D Panov, Spherical surfaces with conical points: systole inequality and moduli spaces with many connected components, Geom. Funct. Anal. 29 (2019) 1110 | DOI

[24] I Pak, Lectures on discrete and polyhedral geometry, (2010)

[25] H P De Saint-Gervais, Uniformization of Riemann surfaces, Eur. Math. Soc. (2016) | DOI

[26] J Song, Y Cheng, B Li, B Xu, Drawing cone spherical metrics via Strebel differentials, Int. Math. Res. Not. 2020 (2020) 3341 | DOI

[27] G Tarantello, Analytical, geometrical and topological aspects of a class of mean field equations on surfaces, Discrete Contin. Dyn. Syst. 28 (2010) 931 | DOI

[28] W P Thurston, Shapes of polyhedra and triangulations of the sphere, from: "The Epstein birthday schrift" (editors I Rivin, C Rourke, C Series), Geom. Topol. Monogr. 1, Geom. Topol. Publ. (1998) 511 | DOI

[29] E T Whittaker, G N Watson, A course of modern analysis, Cambridge Univ. Press (1996) | DOI