## 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

## 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 $X$.## References

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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: katzmik@math.biu.ac.il
**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: sabourau@lmpt.univ-tours.fr
- 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: https://doi.org/10.1090/S0002-9939-05-08057-3
- MathSciNet review: 2196056