Dyck’s surfaces, systoles, and capacities

Katz, Mikhail; Sabourau, Stéphane

doi:10.1090/S0002-9947-2014-06216-8

Dyck’s surfaces, systoles, and capacities
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by Mikhail G. Katz and Stéphane Sabourau PDF

Trans. Amer. Math. Soc. 367 (2015), 4483-4504 Request permission

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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