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Dihedral monodromy of cone spherical metrics. (English) Zbl 1529.30041

Summary: Among metrics of constant positive curvature on a punctured compact Riemann surface with conical singularities at the punctures, dihedral monodromy means that the action of the monodromy group \(\mathcal{M} \subset \operatorname{SO}(3)\) globally preserves a pair of antipodal points. Using recent results about local invariants of quadratic differentials, we give a complete characterization of the set of conical angles realized by some cone spherical metric with dihedral monodromy.

MSC:

30F10 Compact Riemann surfaces and uniformization
30F30 Differentials on Riemann surfaces

References:

[1] A. Eremenko, Co-axial monodromy, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XX (2020), no. 2, 619-634. Digital Object Identifier: 10.2422/2036-2145.201706_022 Google Scholar: Lookup Link MathSciNet: MR4105912 · Zbl 1481.57032 · doi:10.2422/2036-2145.201706_022
[2] A. Eremenko, Metrics of constant positive curvature with four conic singularities on the sphere: A survey, preprint, arXiv:2103.13364 [math.DG]. Digital Object Identifier: 10.48550/arXiv.2103.13364 Google Scholar: Lookup Link · Zbl 1448.34161 · doi:10.48550/arXiv.2103.13364
[3] A. Eremenko, A. Gabrielov, and V. Tarasov, Metrics with conic singularities and spherical polygons, Illinois J. Math. 58 (2014), no. 3, 739-755. zbMATH: 1405.30005 MathSciNet: MR3395961 · Zbl 1405.30005
[4] A. Eremenko, G. Mondello, and D. Panov, Moduli of spherical tori with one conical point, arXiv:2008.02772. Digital Object Identifier: 10.1007/s00039-019-00506-3 Google Scholar: Lookup Link MathSciNet: MR3990195 · Zbl 1447.58013 · doi:10.1007/s00039-019-00506-3
[5] G. Faraco and S. Gupta, Monodromy of Schwarzian equations with regular singularities, preprint, arXiv:2109.04044.
[6] Q. Gendron and G. Tahar, Abelian differentials with prescribed singularities, J. Éc. polytech. Math. 8 (2021), 1397-1428. Digital Object Identifier: 10.1109/jas.2021.1004057 Google Scholar: Lookup Link zbMATH: 1480.30032 MathSciNet: MR4296497 · Zbl 1480.30032 · doi:10.1109/jas.2021.1004057
[7] Q. Gendron and G. Tahar, Quadratic differentials with prescribed singularities, preprint, arXiv:2111.12653. Digital Object Identifier: 10.1109/jas.2021.1004057 Google Scholar: Lookup Link MathSciNet: MR4296497 · Zbl 1480.30032 · doi:10.1109/jas.2021.1004057
[8] M. Heins, On a class of conformal metrics, Nagoya Math. J. 21 (1962), no. 1, 1-60. MathSciNet: MR0143901 · Zbl 0113.05603
[9] E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 1-56. Digital Object Identifier: 10.24033/asens.2062 Google Scholar: Lookup Link zbMATH: 1161.30033 MathSciNet: MR2423309 · Zbl 1161.30033 · doi:10.24033/asens.2062
[10] L. Li, J. Song, and B. Xu, Irreducible cone spherical metrics and stable extensions of two line bundles, Adv. Math. 388 (2021), Paper No. 107854. Digital Object Identifier: 10.1016/j.aim.2021.107854 Google Scholar: Lookup Link MathSciNet: MR4283757 · Zbl 1469.30096 · doi:10.1016/j.aim.2021.107854
[11] R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222-224. Digital Object Identifier: 10.2307/2047555 Google Scholar: Lookup Link zbMATH: 0657.30033 MathSciNet: MR0938672 · Zbl 0657.30033 · doi:10.2307/2047555
[12] G. Mondello and D. Panov, Spherical metrics with conical singularities on a 2-sphere: Angle constraints, Int. Math. Res. Not. IMRN 2016, no. 16, 4937-4995. Digital Object Identifier: 10.1093/imrn/rnv300 Google Scholar: Lookup Link zbMATH: 1446.53027 MathSciNet: MR3556430 · Zbl 1446.53027 · doi:10.1093/imrn/rnv300
[13] J. Song, Y. Cheng, B. Li, and B. Xu, Drawing cone spherical metrics via Strebel differentials, Int. Math. Res. Not. IMRN 2020, no. 11, 3341-3363. Digital Object Identifier: 10.1093/imrn/rny103 Google Scholar: Lookup Link zbMATH: 1487.53056 MathSciNet: MR4123106 · Zbl 1487.53056 · doi:10.1093/imrn/rny103
[14] K. Strebel. Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 5, Springer-Verlag, Berlin, 1984. Digital Object Identifier: 10.1007/978-3-662-02414-0 Google Scholar: Lookup Link MathSciNet: MR0743423 · Zbl 0547.30001 · doi:10.1007/978-3-662-02414-0
[15] G. Tahar, Counting saddle connections in flat surfaces with poles of higher order, Geom. Dedicata 196 (2018), no. 1, 145-186. Digital Object Identifier: 10.1007/s10711-017-0313-2 Google Scholar: Lookup Link zbMATH: 1403.32003 MathSciNet: MR3853632 · Zbl 1403.32003 · doi:10.1007/s10711-017-0313-2
[16] M. Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2) 32 (1986), nos. 1-2, 79-94. MathSciNet: MR0850552 · Zbl 0611.53035
[17] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793-821. Digital Object Identifier: 10.2307/2001742 Google Scholar: Lookup Link zbMATH: 0724.53023 MathSciNet: MR1005085 · Zbl 0724.53023 · doi:10.2307/2001742
[18] A. Zorich, “Flat surfaces” in Frontiers in Number Theory, Physics, and Geometry I, Springer, Berlin, 2006, 437-583. Digital Object Identifier: 10.1007/978-3-540-30308-4 Google Scholar: Lookup Link MathSciNet: MR2290757 · Zbl 1104.11003 · doi:10.1007/978-3-540-30308-4
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