Dyck’s surfaces, systoles, and capacities

Mikhail G. Katz; Stéphane Sabourau

doi:10.1090/s0002-9947-2014-06216-8

Dyck’s surfaces, systoles, and capacities

Mikhail G. Katz, Stéphane Sabourau

Department of Mathematics

Université Paris-Est Créteil

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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 ϑ and two of angle 9π−3ϑ for a suitable ϑ with cos(ϑ) ∈ Q(√ 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.

Original language	English
Pages (from-to)	4483-4504
Number of pages	22
Journal	Transactions of the American Mathematical Society
Volume	367
Issue number	6
DOIs	https://doi.org/10.1090/s0002-9947-2014-06216-8
State	Published - 2015

Bibliographical note

Publisher Copyright:
©2015 American Mathematical Society.

Keywords

Antiholomorphic involution
Capacity
Conformal structure
Dyck’s surface
Extremal metric
Hyperellipticity
Nonpositively curved surface
Optimal systolic inequality
Riemann surface
Systole

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abstract = "We prove an optimal systolic inequality for nonpositively curved Dyck{\textquoteright}s surfaces. The extremal surface is flat with eight conical singularities, six of angle ϑ and two of angle 9π−3ϑ for a suitable ϑ with cos(ϑ) ∈ Q(√ 19). Relying on some delicate capacity estimates, we also show that the extremal surface is not conformally equivalent to the hyperbolic Dyck{\textquoteright}s surface with maximal systole, yielding a first example of systolic extremality with this behavior.",

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AB - We prove an optimal systolic inequality for nonpositively curved Dyck’s surfaces. The extremal surface is flat with eight conical singularities, six of angle ϑ and two of angle 9π−3ϑ for a suitable ϑ with cos(ϑ) ∈ Q(√ 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.

KW - Antiholomorphic involution

KW - Capacity

KW - Conformal structure

KW - Dyck’s surface

KW - Extremal metric

KW - Hyperellipticity

KW - Nonpositively curved surface

KW - Optimal systolic inequality

KW - Riemann surface

KW - Systole

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Dyck’s surfaces, systoles, and capacities

Abstract

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Keywords

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