Low dimensional geometric dynamics

Sharp systolic inequalities for geodesic flows on the two-sphere

speaker: Umberto Hryniewicz (Universidade Federal do Rio de Janeiro)

abstract: Given a Riemannian metric on the two-sphere, consider the systolic ratio, defined as the ratio between the square of the length of the shortest closed geodesic and the total area. In this talk I would like to explain how techniques from symplectic dynamics can be combined with tools from Riemannian geometry to confirm the following conjecture by Babenko and Balacheff: the metric of constant curvature on the two-sphere is a local maximizer of the systolic ratio. In fact, if the metric is \(\delta\)-pinched for some \(\delta>(4+\sqrt{7})/8\) then the systolic ratio is at most \(\pi\), with equality if, and only if, the metric is Zoll. I will also discuss sharp systolic inequalities for Reeb flows on the tight three-sphere close to the standard Hopf fibration. These are results obtained in collaboration with Alberto Abbondandolo, Barney Bramham and Pedro A. S. Salomao.

timetable:
Thu 23 Apr, 9:30 - 10:30, Aula Dini
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