Examples of symplectic capacities

In this section we give the formal definition of symplectic capacities, and discuss a number of examples along with sample applications.

1. Definition. Denote by Symp2n the category of all symplectic manifolds of dimension 2n, with symplectic embeddings as morphisms. A symplectic category is a subcategory 9 of Symp2n such that (M, ω) 2 9 implies (M, αω) 2 9 for all α > 0.

   Convention. We will use the symbol Œ to denote symplectic embeddings and

   ! to denote morphisms in the category 9 (which may be more restrictive).

   D ×

   Let B2n(r 2) be the open ball of radius r in R2n and Z2n(r 2) B2(r 2) R2n—2 the open cylinder (the reason for this notation will become apparent below). Unless stated otherwise, open subsets of R2n are always equipped with the canon-

   j =1

   ical symplectic form ω0 D Pn dyj ^ dxj . We will suppress the dimension

   2n when it is clear from the context and abbreviate

   
   

   B WD B2n(1), Z WD Z2n(1).

   ⊂

   Now let 9 Symp2n be a symplectic category containing the ball B and the cylinder Z. A symplectic capacity on 9 is a covariant functor c from 9 to the category ([0, 1], ≤) (with a ≤ b as morphisms) satisfying

   ≤ !

   (MONOTONICITY): c(M, ω) c(M r, ωr) if there exists a morphism (M, ω) (M r, ωr);

   (CONFORMALITY): c(M, αω) D α c(M, ω) for α > 0; (NONTRIVIALITY): 0 < c(B) and c(Z)< 1.

   Note that the (Monotonicity) axiom just states the functoriality of c. A symplectic capacity is said to be normalized if

   (Normalization): c(B) D 1.

   ⊂

   As a frequent example we will use the set Op2n of open subsets in R2n. We make it into a symplectic category by identifying (U, α2ω0) with the symplectomorphic manifold (αU, ω0) for U R2n and α > 0. We agree that the morphisms in this category shall be symplectic embeddings induced by global symplectomorphisms of R2n. With this identification, the (Conformality) axiom above takes the form

   (CONFORMALITY) r: c(αU ) D α2c(U ) for U 2 Op2n, α > 0.

2. Examples of symplectic capacities

   In this section we give the formal definition of symplectic capacities, and discuss a number of examples along with sample applications.

   1. Definition. Denote by Symp2n the category of all symplectic manifolds of dimension 2n, with symplectic embeddings as morphisms. A symplectic category is a subcategory 9 of Symp2n such that (M, ω) 2 9 implies (M, αω) 2 9 for all α > 0.

      Convention. We will use the symbol Œ to denote symplectic embeddings and

      ! to denote morphisms in the category 9 (which may be more restrictive).

      D ×

      Let B2n(r 2) be the open ball of radius r in R2n and Z2n(r 2) B2(r 2) R2n—2 the open cylinder (the reason for this notation will become apparent below). Unless stated otherwise, open subsets of R2n are always equipped with the canon-

      j =1

      ical symplectic form ω0 D Pn dyj ^ dxj . We will suppress the dimension

      2n when it is clear from the context and abbreviate

      
      

      B WD B2n(1), Z WD Z2n(1).

      ⊂

      Now let 9 Symp2n be a symplectic category containing the ball B and the cylinder Z. A symplectic capacity on 9 is a covariant functor c from 9 to the category ([0, 1], ≤) (with a ≤ b as morphisms) satisfying