Metric and isoperimetric problems in symplectic geometry
Author: Claude Viterbo
Journal: J. Amer. Math. Soc. 13 (2000), 411-431
MSC (1991): Primary 53C15; Secondary 58F05, 49Q99, 58F22, 58E10
Published electronically: January 31, 2000
MathSciNet review: 1750956
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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 , where 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 norm are included.