Hyperelliptic surfaces are Loewner

2005/07/20 by Mikhail G. Katz, Stéphane Sabourau · 2 citations
Mathematics · #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Finite Group Theory Research #Mathematics #Geology

paper · pdf · doi:10.1090/s0002-9939-05-08057-3

openalex publication_date 2005/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

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.

Citations

• Metric Structures for Riemannian and Non-Riemannian Spaces
• Algebraic Curves and Riemann Surfaces
• Filling Riemannian manifolds
• Systolic volume and minimal entropy of aspherical manifolds

Cited by

• Entropy of systolically extremal surfaces and asymptotic bounds
• An optimal systolic inequality for CAT(0) metrics in genus two
• Loewner's torus inequality [wikipedia]