Article

Israel Journal of Mathematics

, Volume 132, Issue 1, pp 221-238

First online:

Genus two extremal surfaces: Extremal discs, isometries and Weierstrass points

  • Ernesto GirondoAffiliated withDepartamento de Matematicas C. Universitaria de Cantoblanco, Universidad Autonoma de Madrid Email author 
  • , Gabino González-DiezAffiliated withDepartamento de Matematicas C. Universitaria de Cantoblanco, Universidad Autonoma de Madrid

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Abstract

It is known that the largest disc that a compact hyperbolic surface of genusg may contain has radiusR=cosh−1(1/2sin(π/(12g−6))). It is also known that the number of such (extremal) surfaces, although finite, grows exponentially withg. Elsewhere the authors have shown that for genusg>3 extremal surfaces contain only one extremal disc.

Here we describe in full detail the situation in genus 2. Following results that go back to Fricke and Klein we first show that there are exactly nine different extremal surfaces. Then we proceed to locate the various extremal discs that each of these surfaces possesses as well as their set of Weierstrass points and group of isometries.