Quantitative symplectic geometry

Abstract

A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux's Theorem such a manifold looks locally like an open set in some R 2n ∼ = C n with the standard symplectic formand so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities.Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer [19,20] (although the first capacity was in fact constructed by M. Gromov [40]). Since then, lots of new capacities have been defined [16,30,32,44,49,59,60,90,99] and they were further studied in [1,2,8,9,17,26,21,28,31,35,37,38,41,42,43,46,48,50,52,56,57,58,61,62,63,64,65,66,68,74,75,76,88,89,91,92,94,97,98]. Surveys on symplectic capacities are [45,50,55,69,97]. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in § 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In § 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. In § 4, we describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion we mention many open problems.As illustrated below, many of the quantitative aspects of symplectic geometry can be formulated in terms of symplectic capacities. Of course there are other numerical invariants of symplectic manifolds which could be included in *

The embedding capacity of 4-dimensional symplectic ellipsoids

“…To date, most results have concerned the embeddings of balls or of products of balls since these are most amenable to analysis. (See Cieliebak, Hofer, Schlenk and Latschev [5] for a comprehensive survey of embedding problems.) However, ellipsoids are another very natural class of examples.…”

“…This is one of a range of symplectic capacity functions defined by Cieliebak, Hofer, Latschev and Schlenk in [5], and is the first to be calculated.…”

The $M$-ellipsoid, symplectic capacities and volume

“…Nowadays, a variety of symplectic capacities can be constructed in different ways. For several of the detailed discussions on symplectic capacities we refer the reader to [2], [7], [8], [10], [12] and [21].…”

Recent progress on symplectic embedding problems in four dimensions

“…However, the EkelandHofer capacities (see ref. 10) show that Pð1;1Þ does not symplectically embed into Eða;2aÞ whenever a < 3∕2. And the latter bound is sharp because, according to our definitions, Pð1;1Þ is a subset of Eð3∕2;3Þ.…”

“…Proof: Copy the above proof of Theorem 1, using Proposition 10 and [11] in place of Proposition 7 and [10].…”