Kodai Mathematical Journal

Extremal disks and extremal surfaces of genus three

Gou Nakamura

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A compact Riemann surface of genus g ≥ 2 is said to be extremal if it admits an extremal disk, a disk of the maximal radius determined by g. If g = 2 or g ≥ 4, it is known that how many extremal disks an extremal surface of genus g can admit. In the present paper we deal with the case of g = 3. Considering the side-pairing patterns of the fundamental polygons, we show that extremal surfaces of genus 3 admit at most two extremal disks and that 16 surfaces admit exactly two. Also we describe the group of automorphisms and hyperelliptic surfaces.

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Kodai Math. J. Volume 28, Number 1 (2005), 111-130.

First available in Project Euclid: 23 March 2005

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Nakamura, Gou. Extremal disks and extremal surfaces of genus three. Kodai Math. J. 28 (2005), no. 1, 111--130. doi:10.2996/kmj/1111588041. http://projecteuclid.org/euclid.kmj/1111588041.

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