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See detailThe maximum number of systoles for genus two Riemann surfaces with abelian differentials
Judge, Chris; Parlier, Hugo UL

in COMMENTARII MATHEMATICI HELVETICI (2019), 94(2), 399-437

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 ... [more ▼]

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. [less ▲]

Detailed reference viewed: 43 (0 UL)
Full Text
Peer Reviewed
See detailThe maximum number of systoles for genus two Riemann surfaces with abelian differentials
Judge, Chris; Parlier, Hugo UL

E-print/Working paper (2017)

This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus ... [more ▼]

This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus two surface and that the bound is realized by a unique translation surface up to homothety. We also provide sharp upper bounds on the the number of homotopy classes of systoles for a holomorphic $1$-form with a single zero in terms of the genus. [less ▲]

Detailed reference viewed: 107 (7 UL)