Hyperellipticity and systoles of Klein surfaces

Mikhail G. Katz, Stéphane Sabourau

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Abstract

Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard's short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov's. A basic idea is to use length bounds for loops on a companion Klein bottle, and then analyze how curves transplant to the original non-orientable surface. We exploit the real structure on the orientable double cover by applying the coarea inequality to the distance function from the real locus. Of particular interest is the case of Dyck's surface. We also exploit an optimal systolic bound for the Möbius band, due to Blatter.

Original languageEnglish
Pages (from-to)277-293
Number of pages17
JournalGeometriae Dedicata
Volume159
Issue number1
DOIs
StatePublished - Aug 2012

Bibliographical note

Funding Information:
Supported by the Israel Science Foundation grant 1294/06.

Keywords

  • Antiholomorphic involution
  • Coarea formula
  • Dyck's surface
  • Hyperelliptic curve
  • Klein bottle
  • Klein surface
  • Loewner's torus inequality
  • Möbius band
  • Riemann surface
  • Systole

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