Displaying similar documents to “Submanifold averaging in Riemannian and symplectic geometry”

Inner and outer hamiltonian capacities

David Hermann (2004)

Bulletin de la Société Mathématique de France

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The aim of this paper is to compare two symplectic capacities in ℂ n related with periodic orbits of Hamiltonian systems: the Floer-Hofer capacity arising from symplectic homology, and the Viterbo capacity based on generating functions. It is shown here that the inner Floer-Hofer capacity is not larger than the Viterbo capacity and that they are equal for open sets with restricted contact type boundary. As an application, we prove that the Viterbo capacity of any compact Lagrangian submanifold...

Characterization of diffeomorphisms that are symplectomorphisms

Stanisław Janeczko, Zbigniew Jelonek (2009)

Fundamenta Mathematicae

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Let ( X , ω X ) and ( Y , ω Y ) be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that Φ * ω Y = c ω X .

On some completions of the space of hamiltonian maps

Vincent Humilière (2008)

Bulletin de la Société Mathématique de France

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In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of ℝ 2 n endowed with the standard symplectic form ω 0 = d p ∧ d q . We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example,...

Submanifolds and the Hofer norm

Michael Usher (2014)

Journal of the European Mathematical Society

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In [Ch00], Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we consider the orbits of more general submanifolds. We show that, for the Chekanov–Hofer pseudometric on the orbit of a closed submanifold to be a genuine metric, it is necessary for the submanifold to be coisotropic, and we show that this...

𝒞 0 -rigidity of characteristics in symplectic geometry

Emmanuel Opshtein (2009)

Annales scientifiques de l'École Normale Supérieure

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The paper concerns a 𝒞 0 -rigidity result for the characteristic foliations in symplectic geometry. A symplectic homeomorphism (in the sense of Eliashberg-Gromov) which preserves a smooth hypersurface also preserves its characteristic foliation.

Fano manifolds of degree ten and EPW sextics

Atanas Iliev, Laurent Manivel (2011)

Annales scientifiques de l'École Normale Supérieure

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O’Grady showed that certain special sextics in ℙ 5 called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that if ( M , ω ) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of  ( M , ω ) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in  M × M ...

Toric structures on near-symplectic 4-manifolds

David T. Gay, Margaret Symington (2009)

Journal of the European Mathematical Society

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A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on...

Robust transitivity in hamiltonian dynamics

Meysam Nassiri, Enrique R. Pujals (2012)

Annales scientifiques de l'École Normale Supérieure

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A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce C r open sets ( r = 1 , 2 , ⋯ , ∞ ) of symplectic diffeomorphisms and Hamiltonian systems, exhibitingrobustly transitive sets. We show that the C ∞ closure of such open sets contains a variety of systems, including so-called unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the...