scholarly journals The Calabi homomorphism, Lagrangian paths and special Lagrangians

2013 ◽  
Vol 357 (4) ◽  
pp. 1389-1424 ◽  
Author(s):  
Jake P. Solomon
Keyword(s):  
Calabi Homomorphism

scholarly journals THE NONLINEAR MASLOV INDEX AND THE CALABI HOMOMORPHISM

2007 ◽  
Vol 09 (06) ◽  
pp. 769-780 ◽  
Author(s):  
GABI BEN SIMON
Keyword(s):  
Projective Space ◽  
Open Subset ◽  
Complex Projective Space ◽  
Maslov Index ◽  
Universal Cover ◽  
Hamiltonian Diffeomorphisms ◽  
Calabi Invariant ◽  
Nonlinear Maslov Index ◽  
Complex Projective ◽  
Calabi Homomorphism

In this paper, we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard (2n - 1)-dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard (n - 1)-dimensional complex projective space. We prove that the value of the pullback of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, when evaluated on a diffeomorphism supported in a sufficiently small open subset of M, equals [Formula: see text] times the Calabi invariant of this diffeomorphism.

The group of symplectomorphisms

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Symplectic Manifold ◽  
Universal Cover ◽  
Topological Properties ◽  
Basic Properties ◽  
Group Of Diffeomorphisms ◽  
Path Connected ◽  
Flux Homomorphism ◽  
Calabi Homomorphism

This chapter discusses the basic properties of the group of symplectomorphisms of a compact connected symplectic manifold and its subgroup of Hamiltonian symplectomorphisms. It begins by showing that the group of symplectomorphisms is locally path-connected and then moves on to the flux homomorphism. The main result here is a theorem of Banyaga that characterizes the Hamiltonian symplectomorphisms in terms of the flux homomorphism. In the noncompact case there is another interesting homomorphism, called the Calabi homomorphism, that takes values in the reals and may be defined on the universal cover of the group of Hamiltonian symplectomorphisms. The chapter ends with a brief comparison of the topological properties of the group of symplectomorphisms with those of the group of diffeomorphisms.

scholarly journals Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk

2021 ◽  
Author(s):  
David Bechara Senior
Keyword(s):  
Integral Formula ◽  
Winding Number ◽  
Compactly Supported ◽  
Area Preserving ◽  
Calabi Homomorphism

Abstract Given a compactly supported area-preserving diffeomorphism of the disk, we prove an integral formula relating the asymptotic action to the asymptotic winding number. As a corollary, we obtain a new proof of Fathi’s integral formula for the Calabi homomorphism on the disk.

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