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The maximum number of systoles for genus two Riemann surfaces with
abelian differentials
release_cffor2dtdfgzbah6d4jd46dhfa
by
Chris Judge, Hugo Parlier
Released
as a article
.
2017
Abstract
In this article, we provide bounds on systoles associated to a holomorphic
1-form ω 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,ω) and, moreover, that this bound is realized by a unique translation
surface up to homothety. For general genus g and a holomorphic 1-form
ω 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.
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1703.01809v2
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