Quantitative symplectic geometry

By Kai Cieliebak, Helmut Hofer, J. Latschev, F. Schlenk. 2005

116 citations recorded by the source corpus.

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 ℝ2n≅ ℂnwith the standard symplectic form and 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 (although the first capacity was in fact constructed by M. Gromov). Since then, lots of new capacities have been defined and they were further studied in. Surveys on symplectic capacities are. 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 Section 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In Section 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b79274c3c4181266e9fed3876bf4d8b9ca18913b

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

The Hofer conjecture on embedding symplectic ellipsoids (2010, result): “Hofer’s conjecture evolved as earlier guesses, such as those by Cieliebak, Hofer, Latschev and Schlenk in [3], proved inadequate.”
The embedding capacity of 4-dimensional symplectic ellipsoids (2009, background): “Moreover, its graph is linear on each subinterval [1, 2], [2, 4], [4, 5], [5, 6].”
Symplectic embeddings of polydisks (2007, unknown): “ in R2n in such a way that if U symplectically embeds in V then c(U) ≤ c(V ). The function c should also scale in a reasonable way if we scale the set U, and it obeys some other mild conditions. (See [6] for all details.) The volume is a trivial example of a generalized capacity. The ﬁrst non-trivia”
Sharp systolic inequalities for Reeb flows on the three-sphere (2015, unknown): “r>s. Many non-equivalent symplectic capacities have been constructed so far, such as the Ekeland-Hofer capacity cEH and the Hofer-Zehnder capacity cHZ (see [EH89, HZ90, HZ94]). See also the survey [CHLS07] for more information on symplectic capacities. Let cbe a symplectic capacity on (R2n,ω 0). ”
Lecture Notes on Embedded Contact Homology (2013, unknown): “) respectively [5, 13].”
Quantitative embedded contact homology (2010, background): “For background on symplectic embedding questions more generally we refer the reader to [3] for an extensive discussion.”
Symplectic embeddings of 4‐dimensional ellipsoids (2008, background): “The multiplicities 2, 1, 3, 2 of the weights are the terms (or partial quotients) of the continued fraction [2; 1, 3, 2] representation of 25/9.”
Punctured holomorphic curves and Lagrangian embeddings (2014, background): “This is the embedding capacity of D∗(Q, g) into (X,ω) in the sense of [16].”
Symplectic embedding problems, old and new (2017, background): “There are by now a dozen of different symplectic capacities that are defined in various ways: by symplectic embedding problems, symplectic areas of J-holomorphic curves, variational problems in Hamiltonian dynamics, or as critical values of generating functions; see the survey [20].”
Erratum to “Geometry of contact transformations and domains: orderability versus squeezing” (2005, background): “For instance, all convex domains lie in ΞD for D = 4n 2 (see [V2],[CHLS]).”