Commentarii Mathematici Helvetici


Full-Text PDF (638 KB) | Metadata | Table of Contents | CMH summary
Volume 94, Issue 2, 2019, pp. 399–437
DOI: 10.4171/CMH/463

Published online: 2019-04-17

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

Chris Judge[1] and Hugo Parlier[2]

(1) Indiana University, Bloomington, USA
(2) University of Luxembourg, Esch-sur-Alzette, Luxembourg

In this article, we provide bounds on systoles associated to a holomorphic 1-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X, \omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $\omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.

Keywords: Systoles, translation surfaces, abelian differentials

Judge Chris, Parlier Hugo: The maximum number of systoles for genus two Riemann surfaces with abelian differentials. Comment. Math. Helv. 94 (2019), 399-437. doi: 10.4171/CMH/463