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Title:
Positive loops---on a question by Eliashberg-Polterovich and a contact systolic inequality
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Abstract:
In 2000 Eliashberg-Polterovich introduced the concept of positivity in contact geometry. The notion of a positive loop of contactomorphisms is central. A question of Eliashberg-Polterovich is whether $C^0$-small positive loops exist. We give a negative answer to this question. Moreover we give sharp lower bounds for the size which, in turn, gives rise to a $L^\infty$-contact systolic inequality. This should be contrasted with a recent result by Abbondandolo et. al. that on the standard contact 3-sphere no $L^2$-contact systolic inequality exists. The choice of $L^2$ is motivated by systolic inequalities in Riemannian geometry. This is joint work with U. Fuchs and W. Merry.
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