Metric and isoperimetric problems in symplectic geometry

Viterbo, Claude¹

¹Département de Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay Cedex, France

Journal of the American Mathematical Society, Tome 13 (2000) no. 2, pp. 411-431

Résumé

Our first result is a reduction inequality for the displacement energy. We apply it to establish some new results relating symplectic capacities and the volume of a Lagrangian submanifold in a number of different settings. In particular, we prove that a Lagrange submanifold always bounds a holomorphic disc of area less than $C_{n}\operatorname {vol}(L)^{2/n}$, where $C_{n}$ is some universal constant. We also explain how the Alexandroff-Bakelman-Pucci inequality is a special case of the above inequalities. Our inequality on displacement of reductions is also applied to yield a relation between length of billiard trajectories and volume of the domain. Two simple results concerning isoperimetric inequalities for convex domains and the closure of the symplectic group for the $W^{1/2,2}$ norm are included.

Détail

DOI : 10.1090/S0894-0347-00-00328-3

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