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Conformal Geometry and Dynamics

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Compact non-orientable surfaces of genus 6 with extremal metric discs


Author: Gou Nakamura
Journal: Conform. Geom. Dyn. 20 (2016), 218-234
MSC (2010): Primary 30F50; Secondary 30F40, 05C10
DOI: https://doi.org/10.1090/ecgd/298
Published electronically: June 20, 2016
MathSciNet review: 3513567
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Abstract: A compact hyperbolic surface of genus $ g$ is said to be extremal if it admits an extremal disc, a disc of the largest radius determined only by $ g$. We discuss how many extremal discs are embedded in non-orientable extremal surfaces of genus 6. This is the final genus in our interest because it is already known for $ g=3, 4, 5$, or $ g>6$. We show that non-orientable extremal surfaces of genus 6 admit at most two extremal discs. The locus of extremal discs is also obtained for each surface. Consequently non-orientable extremal surfaces of arbitrary genus $ g\geqq 3$ admit at most two extremal discs. Furthermore we determine the groups of automorphisms of non-orientable extremal surfaces of genus 6 with two extremal discs.


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Additional Information

Gou Nakamura
Affiliation: Science Division, Center for General Education, Aichi Institute of Technology, Yakusa-Cho Toyota 470-0392, Japan
Email: gou@aitech.ac.jp

DOI: https://doi.org/10.1090/ecgd/298
Received by editor(s): October 10, 2015
Received by editor(s) in revised form: February 17, 2016
Published electronically: June 20, 2016
Additional Notes: This work was partially supported by the JSPS KAKENHI Grant No. 25400147.
Dedicated: Dedicated to Professor Noriaki Suzuki on the occasion of his sixtieth birthday
Article copyright: © Copyright 2016 American Mathematical Society