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Différentielles quadratiques à singularités prescrites

Part of: Riemann surfaces

Published online by Cambridge University Press:  28 May 2024

Quentin Gendron  and
Guillaume Tahar
Quentin Gendron
Affiliation:
Instituto de Matemáticas de la UNAM Ciudad Universitaria, CDMX, 04510 Mexico City, México e-mail: quentin.gendron@im.unam.mx
Guillaume Tahar*
Affiliation:
Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, China
*
e-mail: guillaume.tahar@bimsa.cn
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Abstract

The local invariants of a meromorphic quadratic differential on a compact Riemann surface are the orders of zeros and poles, and the residues at the poles of even orders. The main result of this paper is that with few exceptions, every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface. The exceptions are completely classified and only occur in genera zero and one. Moreover, in the case of a nonconnected stratum, we show that, with three exceptions in genus one, each configuration of invariants can be realized in each non-hyperelliptic connected component of the stratum. In the hyperelliptic components with two poles the residues at both poles coincide. These results are obtained using the flat metric induced by the differentials. We give an application by bounding the number of disjoint cylinders on a primitive quadratic differential.

Keywords

Quadratic differentialflat surfacestrataresidue

MSC classification

Primary: 30F30: Differentials on Riemann surfaces
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 65
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Bainbridge, M., Chen, D., Gendron, Q., Grushevsky, S., and Möller, M., Strata of $k$ -differentials . Algebr. Geom. 6(2019), no. 2, 196–233.Google Scholar
Chen, D. and Gendron, Q., Towards a classification of connected components of the strata of $k$ -differentials . Doc. Math. 27(2022), 1031–1100.Google Scholar
Dias, K., Gupta, S., and Trnkova, M., Quadratic differentials, measured foliations, and metric graphs on punctured surfaces . Ill. J. Math. 65(2021), no. 2, 417–454.Google Scholar
Diaz-Marin, H., Meromorphic quadratic differentials with prescribed singularities . Bol. Soc. Bras. Mat., Nova Sér. 31(2000), no. 2, 189–204.Google Scholar
Gendron, Q. and Tahar, G., Différentielles à singularités prescrites . Preprint, 2017. arxiv:1705.03240.Google Scholar
Gendron, Q. and Tahar, G., Différentielles abéliennes à singularités prescrites . J. del’ École Polytechnique — Math é matiques 8(2021), 1397–1428.Google Scholar
Gendron, Q. and Tahar, G., Dihedral monodromy of cone spherical metrics . Ill. J. Math. 67(2023), no. 3, 457–483.Google Scholar
Gendron, Q. and Tahar, G., $k$ -différentielles à singularités prescrites . Preprint, 2022. arxiv:2208.11654.Google Scholar
Lanneau, E., Connected components of the strata of the moduli spaces of quadratic differentials . Ann. Sci. Éc. Norm. Supér. (4). 41(2008), no. 1, 1–56.Google Scholar
Masur, H. and Smillie, J., Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms . Comment. Math. Helv. 68(1993), no. 2, 289–307.Google Scholar
Naveh, Y., Tight upper bounds on the number of invariant components on translation surfaces . Isr. J. Math. 165(2008), no. 1, 211–231.Google Scholar
Song, J., Cheng, Y., Li, B., and Bin, X., Drawing cone spherical metrics via Strebel differentials . Int. Math. Res. Not. 2020(2020), no. 11, 3341–3363.Google Scholar
Strebel, K.. Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, Band 5, Springer XII, Berlin, Heidelberg: Springer, 1984.Google Scholar
Tahar, G., Counting saddle connections in flat surfaces with poles of higher order . Geom. Dedicata. 196(2018), no. 1, 145–186.Google Scholar