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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hyperelliptic surfaces are Loewner
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by Mikhail G. Katz and Stéphane Sabourau PDF
Proc. Amer. Math. Soc. 134 (2006), 1189-1195 Request permission


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
  • MR Author ID: 197211
  • Email:
  • 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
  • Email:
  • 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
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 1189-1195
  • MSC (2000): Primary 53C23; Secondary 30F10
  • DOI:
  • MathSciNet review: 2196056