THE NONLINEAR MASLOV INDEX AND THE 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.

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CALABI QUASIMORPHISMS FOR THE SYMPLECTIC BALL

Communications in Contemporary Mathematics ◽

10.1142/s0219199704001525 ◽

2004 ◽

Vol 06 (05) ◽

pp. 793-802 ◽

Cited By ~ 37

Author(s):

PAUL BIRAN ◽

MICHAEL ENTOV ◽

LEONID POLTEROVICH

Keyword(s):

Projective Space ◽

Open Subset ◽

Complex Projective Space ◽

Clifford Torus ◽

Compactly Supported ◽

Linearly Independent ◽

Calabi Invariant ◽

Hofer Metric ◽

Complex Projective

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show that the (Lagrangian) Clifford torus in the complex projective space cannot be displaced from itself by a Hamiltonian isotopy.

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REMARKS ON INVARIANTS OF HAMILTONIAN LOOPS

Journal of Topology and Analysis ◽

10.1142/s1793525310000367 ◽

2010 ◽

Vol 02 (03) ◽

pp. 277-325 ◽

Cited By ~ 5

Author(s):

EGOR SHELUKHIN

Keyword(s):

Maslov Index ◽

Universal Cover ◽

Symplectic Manifolds ◽

Linear Functions ◽

Torus Action ◽

Hamiltonian Diffeomorphisms ◽

Toric Manifolds ◽

Mixed Action ◽

Nonlinear Maslov Index ◽

Moment Polytopes

In this paper the interrelations between several natural morphisms on the π1 of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (nonlinear) Maslov index of loops of quantomorphisms of prequantizations of ℂPn and the Calabi–Weinstein invariant is shown, settling affirmatively a conjecture by Givental. We also prove, in the wake of a remark by Woodward, the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds. Finally, a family of generalized action-Maslov invariants is computed for toric manifolds, on loops coming from the torus action, via barycenters of their moment polytopes, with an application to mass-linear functions recently introduced by McDuff and Tolman. In addition, we reinterpret the quasimorphism of Py on the universal cover of the group of Hamiltonian diffeomorphisms of monotone symplectic manifolds.

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Non-Compact Complex Projective Space as a Phase Space

Physics of Particles and Nuclei Letters ◽

10.1134/s1547477120050210 ◽

2020 ◽

Vol 17 (5) ◽

pp. 744-747

Author(s):

E. Khastyan ◽

H. Shmavonyan

Keyword(s):

Phase Space ◽

Projective Space ◽

Complex Projective Space ◽

Complex Projective

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Sectional curvatures of homogeneous real hypersurfaces of types (A) and (B) in a complex projective space

Journal of Geometry ◽

10.1007/s00022-021-00585-4 ◽

2021 ◽

Vol 112 (2) ◽

Author(s):

Makoto Kimura ◽

Sadahiro Maeda ◽

Hiromasa Tanabe

Keyword(s):

Projective Space ◽

Complex Projective Space ◽

Real Hypersurfaces ◽

Sectional Curvatures ◽

Complex Projective

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From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700040302 ◽

2002 ◽

Vol 66 (3) ◽

pp. 465-475 ◽

Cited By ~ 1

Author(s):

J. Bolton ◽

C. Scharlach ◽

L. Vrancken

Keyword(s):

Projective Space ◽

Minimal Surface ◽

Complex Projective Space ◽

Lagrangian Submanifold ◽

3 Dimensional ◽

Ellipse Of Curvature ◽

Complex Projective

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

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