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 C0-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∞-contact systolic inequality. This should be contrasted with a recent result by Abbondandolo et. al. that on the standard contact 3-sphere no L2-contact systolic inequality exists. The choice of L2 is motivated by systolic inequalities in Riemannian geometry.
This is joint work with U. Fuchs and W. Merry.
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
