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Hyperelliptic surfaces are Loewner

Authors: Mikhail G. Katz and Stéphane Sabourau
Journal: Proc. Amer. Math. Soc. 134 (2006), 1189-1195
MSC (2000): Primary 53C23; Secondary 30F10
Published electronically: July 20, 2005
MathSciNet review: 2196056
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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 $\varepsilon$-regular metrics on $X$.

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Additional Information

Mikhail G. Katz
Affiliation: Department of Mathematics and Statistics, Bar Ilan University, Ramat Gan 52900, Israel

Stéphane Sabourau
Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37400 Tours, France
Address at time of publication: Mathematics and Computer Science Department, St. Joseph’s University, 5600 City Avenue, Philadelphia, Pennsylvania 19131

Keywords: $\varepsilon$-regular metrics, Hermite constant, hyperelliptic involution, Loewner inequality, Pu's inequality, systole, Weierstrass point
Received by editor(s): March 18, 2004
Received by editor(s) in revised form: October 26, 2004
Published electronically: July 20, 2005
Additional Notes: The first author was supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03)
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2005 American Mathematical Society

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