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Dyck's surfaces, systoles, and capacities


Authors: Mikhail G. Katz and Stéphane Sabourau
Journal: Trans. Amer. Math. Soc. 367 (2015), 4483-4504
MSC (2010): Primary 53C23; Secondary 30F10, 58J60
DOI: https://doi.org/10.1090/S0002-9947-2014-06216-8
Published electronically: October 10, 2014
MathSciNet review: 3324936
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Abstract: We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle $ \vartheta $ and two of angle  $ 9 \pi -3 \vartheta $ for a suitable $ \vartheta $ with $ \cos (\vartheta )\in \mathbb{Q}(\sqrt {19})$. Relying on some delicate capacity estimates, we also show that the extremal surface is not conformally equivalent to the hyperbolic Dyck's surface with maximal systole, yielding a first example of systolic extremality with this behavior.


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Additional Information

Mikhail G. Katz
Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan 52900, Israel
Email: katzmik@macs.biu.ac.il

Stéphane Sabourau
Affiliation: Laboratoire d’Analyse et Mathématiques Appliquées, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil, France
Email: stephane.sabourau@u-pec.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-06216-8
Keywords: Systole, optimal systolic inequality, extremal metric, nonpositively curved surface, Riemann surface, Dyck's surface, hyperellipticity, antiholomorphic involution, conformal structure, capacity
Received by editor(s): February 20, 2013
Received by editor(s) in revised form: June 3, 2013, and January 6, 2014
Published electronically: October 10, 2014
Article copyright: © Copyright 2014 American Mathematical Society