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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
DOI: https://doi.org/10.1090/S1088-4173-09-00194-5
Published electronically: April 22, 2009
MathSciNet review: 2497316
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Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • 1. C. Bavard, Disques extrémaux et surfaces modulaires, Ann. de la Fac. des Sciences de Toulouse V (1996), no. 2, 191-202. MR 1413853 (97i:30059)
  • 2. E. Girondo and G. González-Diez, On extremal discs inside compact hyperbolic surfaces, C. R. Acad. Sci. Paris, t. 329, Série I (1999), 57-60. MR 1703263 (2000f:53036)
  • 3. E. Girondo and G. González-Diez, Genus two extremal surfaces: extremal discs, isometries and Weierstrass points, Israel J. Math. 132 (2002), 221-238. MR 1952622 (2003k:57020)
  • 4. E. Girondo and G. Nakamura, Compact non-orientable hyperbolic surfaces with an extremal metric disc, Conform. Geom. Dyn. 11 (2007), 29-43. MR 2295996 (2007k:30087)
  • 5. T. Jørgensen and M. Näätänen, Surfaces of genus 2: generic fundamental polygons, Quart. J. Math. Oxford (2), 33 (1982), 451-461. MR 679814 (84c:51029)
  • 6. G. Nakamura, The number of extremal disks embedded in compact Riemann surfaces of genus two, Sci. Math. Japon 56 (2002), no. 3, 481-492. MR 1937911 (2003i:30065)
  • 7. G. Nakamura, Extremal disks and extremal surfaces of genus three, Kodai Math. J. 28 (2005), no. 1, 111-130. MR 2122195 (2005j:30055)

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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/S1088-4173-09-00194-5
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.

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