Symplectic capacity

In symplectic Geometry, the symplectic capacities are classes of Invariant S checking of the properties of monotonicity and conformity, formalized following work of Ekeland, Hofer and Zehnder.

Definition

A symplectic capacity is the data for any symplectic variety ( M , $\ omega$ ) of a positive real number or the infinite one, noted C ( M , $\ omega$ ), checking conditions of growth, homogeneity, and not-commonplace:

Growth: If there exists a plunging symplectic of ( M , $\ omega$ ) in ( NR , $\ omega'$ ), then:
: $C (M, \ Omega) \ Leq C (N, \ omega')$ .
Homogeneity: For any reality R not no one, one a:
: $C (M, R. \ Omega) \ Leq |R|. C (M, \ Omega)$ .
Not-commonplace: the symplectic capacity of the two open ones of C ^N is finished and not no one:
: $B= \ {Z \ in C^n, |Z|<1\}$
: $Z=S^1\times C^{n-1}$ .

There is not unicity of the symplectic capacities. There exist various approaches to define some explicitly.

Application

In 1981, Eliashberg shows that the group of the symplectomorphisms of a compact symplectic variety is closed for topology C ⁰. In 1990, Ekeland and Hofer provide a demonstration calling upon the use of the symplectic capacities. The diagram of the demonstration is the following. A symplectic capacity induced a real application on the whole of the open parts of a compact symplectic variety ( M ,

\ omega

), continuous for the distance from Hausdorff. A homeomorphism of M preserves this application if it is a diffeomorphism symplectic or antisymplectic. Continuity implies that the group Symp ( M ,

\ omega

) is closed for the topology of uniform convergence. Equivalence is the heart of the demonstration.

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