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

2017/03/06 by Judge, Chris, Parlier, Hugo · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1703.01809

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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• Systolic embedding of graphs on translation surfaces

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• Translation surfaces with large systoles
• The homology systole of hyperbolic Riemann surfaces
• Construction of surfaces with large systolic ratio
• Construction of hyperbolic Riemann surfaces with large systoles
• Trace Systoles and Sink Constant