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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 ▲]

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See detailLorentzian manifolds with a conformal action of SL(2,R)
Pecastaing, Vincent UL

in Commentarii Mathematici Helvetici (2018), 98(2), 401-439

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the ... [more ▼]

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat - that is, everywhere locally conformally diffeomorphic to Minkowski space-time. When the group is non-compact and not locally isomorphic to SO(1,n), n>1, we derive global conclusions, extending a theorem of Frances and Zeghib to some simple Lie groups of real-rank 1. This result is also a first step towards a classification of the conformal groups of compact Lorentz manifolds, analogous to a classification of their isometry groups due to Adams, Stuck and, independently, Zeghib at the end of the 1990's. [less ▲]

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