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

Bar-Ilan University

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

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

Keywords

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

All Science Journal Classification (ASJC) codes

Applied Mathematics
General Mathematics

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10.1090/s0002-9947-2014-06216-8

Cite this

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title = "Dyck{\textquoteright}s surfaces, systoles, and capacities",

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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doi = "10.1090/s0002-9947-2014-06216-8",

language = "الإنجليزيّة",

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TY - JOUR

T1 - Dyck’s surfaces, systoles, and capacities

AU - Katz, Mikhail G.

AU - Sabourau, Stéphane

PY - 2015

Y1 - 2015

N2 - 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.

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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U2 - 10.1090/s0002-9947-2014-06216-8

DO - 10.1090/s0002-9947-2014-06216-8

M3 - مقالة

SN - 0002-9947

VL - 367

SP - 4483

EP - 4504

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -

Dyck’s surfaces, systoles, and capacities

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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