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Conformal Geometry and Dynamics

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Compact non-orientable surfaces of genus $4$ with extremal metric discs

Author: Gou Nakamura
Journal: Conform. Geom. Dyn. 13 (2009), 124-135
MSC (2000): Primary 30F50; Secondary 30F40
Published electronically: April 22, 2009
MathSciNet review: 2497316
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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 by $g$. We know how many extremal discs are embedded in a non-orientable extremal surface of genus $g=3$ or $g>6$. We show in the present paper that there exist $144$ non-orientable extremal surfaces of genus $4$, and find the locations of all extremal discs in those surfaces. As a result, each surface contains at most two extremal discs. Our methods used here are similar to those in the case of $g=3$.

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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
MR Author ID: 639802

Keywords: Extremal discs, Klein surfaces
Received by editor(s): March 27, 2008
Published electronically: April 22, 2009
Additional Notes: This work was supported in part by Grant-in-Aid for Young Scientists (B) (No. 20740081).
Dedicated: Dedicated to Professor Yoshihiro Mizuta on the occasion of his 60th birthday
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.