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Curiosités Lagrangiennes en dimension 4

Denis Sauvaget (2004)

Annales de l’institut Fourier

Dans ce texte, on définit, pour les immersions lagrangiennes de variétés fermées dans n , une notion d’aire symplectique enlacée. Puis on construit, dans le cas n = 2 , un certain nombre de surfaces lagrangiennes enlaçant une aire infinie. Dans le cas des surfaces exactes, elles ont le minimum de points doubles possible permis par la théorie (sauf la sphère), c’est-à-dire moins que prévu par quelques conjectures.

General spectral flow formula for fixed maximal domain

Bernhelm Booss-Bavnbek, Chaofeng Zhu (2005)

Open Mathematics

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by...

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space ( W , Ω ¯ ) , having chosen a given reference...

Global finite generating functions for field theory

Franco Cardin (2003)

Banach Center Publications

We introduce an infinite-dimensional version of the Amann-Conley-Zehnder reduction for a class of boundary problems related to nonlinear perturbed elliptic operators with symmetric derivative. We construct global generating functions with finite auxiliary parameters, describing the solutions as critical points in a finite-dimensional space.

Graph selectors and viscosity solutions on Lagrangian manifolds

David McCaffrey (2006)

ESAIM: Control, Optimisation and Calculus of Variations

Let Λ be a Lagrangian submanifold of T * X for some closed manifold X. Let S ( x , ξ ) be a generating function for Λ which is quadratic at infinity, and let W(x) be the corresponding graph selector for Λ , in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset X 0 X of measure zero such that W is Lipschitz continuous on X, smooth on X X 0 and ( x , W / x ( x ) ) Λ for X X 0 . Let H(x,p)=0 for ( x , p ) Λ . Then W is a classical solution to H ( x , W / x ( x ) ) = 0 on X X 0 and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational...

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

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 when M satisfies...

Invariance of global solutions of the Hamilton-Jacobi equation

Ezequiel Maderna (2002)

Bulletin de la Société Mathématique de France

We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.

Lagrangian holonomy ; characteristic elements of a lagrangian foliation

Carlos Currás-Bosch, Pierre Molino (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let be a lagrangian foliation on a symplectic manifold ( M 2 n , ω ) . The characteristic elements of such a foliation associated to a lagrangian total transversal are obtained; they are a generalisation of the characteristic elements given by J.J. Duistermaat [5]. This technique is applied to give a classification of the germs of lagrangian foliation along a compact leaf. Several examples of classification are given.

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