scholarly journals On a local systolic inequality for odd-symplectic forms

2020 ◽  
Vol 76 (3) ◽  
pp. 327-394
Author(s):  
Gabriele Benedetti ◽  
Jungsoo Kang
Keyword(s):  
Symplectic Forms ◽  
Systolic Inequality

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals Fiberwise convexity of Hill’s lunar problem

2017 ◽  
Vol 09 (04) ◽  
pp. 571-630 ◽  
Author(s):  
Junyoung Lee
Keyword(s):  
Energy Level ◽  
Contact Structure ◽  
Finsler Geometry ◽  
Critical Energy ◽  
Critical Value ◽  
Finsler Metrics ◽  
Tight Contact ◽  
Systolic Inequality ◽  
Geodesic Problem ◽  
Hill's Lunar Problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

The Weil-Petersson and Thurston symplectic forms

2001 ◽  
Vol 108 (3) ◽  
pp. 581-597 ◽  
Author(s):  
Yaşar Sözen ◽  
Francis Bonahon
Keyword(s):  
Symplectic Forms

Equivariant Symplectic Forms

1999 ◽  
pp. 111-147
Author(s):  
Victor W. Guillemin ◽  
Shlomo Sternberg ◽  
Jochen Brüning
Keyword(s):  
Symplectic Forms

scholarly journals Unobstructed symplectic packing for tori and hyper-Kähler manifolds

2016 ◽  
Vol 08 (04) ◽  
pp. 589-626 ◽  
Author(s):  
Michael Entov ◽  
Misha Verbitsky
Keyword(s):  
Structure Theorem ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Orbit Closure ◽  
Kahler Manifolds ◽  
Dimensional Torus ◽  
Symplectic Forms ◽  
Kahler Manifold ◽  
Symplectic Packing ◽  
Smooth Deformation

Let [Formula: see text] be a closed symplectic manifold of volume [Formula: see text]. We say that the symplectic packings of [Formula: see text] by balls are unobstructed if any collection of disjoint symplectic balls (of possibly different radii) of total volume less than [Formula: see text] admits a symplectic embedding to [Formula: see text]. In 1994, McDuff and Polterovich proved that symplectic packings of Kähler manifolds by balls can be characterized in terms of the Kähler cones of their blow-ups. When [Formula: see text] is a Kähler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple), these Kähler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that for any Campana simple Kähler manifold, as well as for any manifold which is a limit of Campana simple manifolds in a smooth deformation, the symplectic packings by balls are unobstructed. This is used to show that the symplectic packings by balls of all even-dimensional tori equipped with Kähler symplectic forms and of all hyper-Kähler manifolds of maximal holonomy are unobstructed. This generalizes a previous result by Latschev–McDuff–Schlenk. We also consider symplectic packings by other shapes and show, using Ratner’s orbit closure theorem, that any even-dimensional torus equipped with a Kähler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).

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