Abstract
The purpose of this note is to study the π1 andH 1 systoles of two Riemann surfaces, the Klein genus 3 surface, y7 =x 2(1 −x), which we will denoteS K , and Bolza’s surface of genus 2,y 2 =x(1 −x 4), which we will denoteS B . Using period matrices, we identify theH 1 systoles and show for these examples that the π1 andH 1 systoles are in one-to-one correspondence. In their respective Teichmüller spaces,S B is known to be extremal andS K locally extremal for these systoles.
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Quine, J.R. Systoles of two extremal Riemann surfaces. J Geom Anal 6, 461–474 (1996). https://doi.org/10.1007/BF02921661
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DOI: https://doi.org/10.1007/BF02921661