The maximum number of systoles for genus two Riemann surfaces with abelian differentials

  • Chris Judge

    Indiana University, Bloomington, USA
  • Hugo Parlier

    University of Luxembourg, Esch-sur-Alzette, Luxembourg
The maximum number of systoles for genus two Riemann surfaces with abelian differentials cover

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Abstract

In this article, we provide bounds on systoles associated to a holomorphic 1-form on a Riemann surface . In particular, we show that if has genus two, then, up to homotopy, there are at most systolic loops on and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus and a holomorphic 1-form with one zero, we provide the optimal upper bound, , on the number of homotopy classes of systoles. If, in addition, is hyperelliptic, then we prove that the optimal upper bound is .

Cite this article

Chris Judge, Hugo Parlier, The maximum number of systoles for genus two Riemann surfaces with abelian differentials. Comment. Math. Helv. 94 (2019), no. 2, pp. 399–437

DOI 10.4171/CMH/463