Hyperelliptic surfaces are loewner

Mikhail G. Katz; Stéphane Sabourau

doi:10.1090/S0002-9939-05-08057-3

Hyperelliptic surfaces are loewner

Mikhail G. Katz
, Stéphane Sabourau

Department of Mathematics

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

22 Scopus citations

Abstract

We prove that C. Loewner's inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces X as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on X. In higher genus, we exploit M. Gromov's area estimates for ε-regular metrics on X.

Original language	English
Pages (from-to)	1189-1195
Number of pages	7
Journal	Proceedings of the American Mathematical Society
Volume	134
Issue number	4
DOIs	https://doi.org/10.1090/S0002-9939-05-08057-3
State	Published - Apr 2006

Keywords

Hermite constant
Hyperelliptic involution
Loewner inequality
Pu's inequality
Systole
Weierstrass point
ε-regular metrics

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10.1090/S0002-9939-05-08057-3

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title = "Hyperelliptic surfaces are loewner",

abstract = "We prove that C. Loewner's inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces X as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on X. In higher genus, we exploit M. Gromov's area estimates for ε-regular metrics on X.",

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AU - Sabourau, Stéphane

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N2 - We prove that C. Loewner's inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces X as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on X. In higher genus, we exploit M. Gromov's area estimates for ε-regular metrics on X.

AB - We prove that C. Loewner's inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces X as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on X. In higher genus, we exploit M. Gromov's area estimates for ε-regular metrics on X.

KW - Hermite constant

KW - Hyperelliptic involution

KW - Loewner inequality

KW - Pu's inequality

KW - Systole

KW - Weierstrass point

KW - ε-regular metrics

UR - https://www.scopus.com/pages/publications/33645797078

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EP - 1195

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 4

ER -

Hyperelliptic surfaces are loewner

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Keywords

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