$b^m$-symplectic manifolds: symmetries, classification and stability

• Autores: Arnau Planas Bahí
• Directores de la Tesis: Eva Miranda Galcerán (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2020
• Idioma: español
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• Resumen

  • This thesis explores classification and perturbation problems for group actions on a class of Poisson manifolds called $b^m$-Poisson manifolds.

    $b^m$-Poisson manifolds are manifolds which are symplectic away from a hypersurface along which they satisfy some transversality properties. They often model problems on symplectic manifolds with boundary such as the study of their deformation quantization and celestial mechanics.

    One of the interesting properties of $b^m$-Poisson manifolds is that their study can be achieved considering the language of $b^m$-forms. That is to say, we can work with forms which are symplectic away from the critical set and admit a smooth extension as a \emph{form over a Lie algebroid} generalizing De Rham forms as form over the standard Lie algebroid of the tangent bundle of the manifold. To consider $b^m$-forms the standard tangent bundle is replaced by the $b^m$-tangent bundle.

    This thesis starts with the equivariant classification of $b^m$-Poisson structures investigating, in particular, the analogue of Moser's classification theorem for symplectic surfaces and their equivariant analogues. The classification invariants in the case of surfaces are encoded in a cohomology called $b^m$-cohomology which has been deeply studied by \cite{Scott16}. Mazzeo-Melrose type formula for $b^m$-cohomology decomposes it in two pieces which can be read off the De Rham cohomology of both the ambient manifold $M$ and the critical hypersurface. As an outcome of this identification, the Poisson classification of these manifolds is given by the De Rham cohomology of the manifold and the hypersurface.

    This classification is extended to the equivariant setting if we assume that the singular forms are preserved by the group action of a compact Lie group.

    These techniques can be extended to the classification of $b^m$-Nambu structures which are also considered in this thesis.

    Group actions re-appear in the last chapters as integrable systems on these manifolds turn out to have associated Hamiltonian actions of tori in a neighbourhood of a Liouville torus. We use this Hamiltonian group action to prove existence of action-angle coordinates in a neighborhood of a Liouville torus. The action-angle coordinate theorem that we prove gives a semilocal normal form in the neighbourhood of a Liouville torus for the $b^m$-symplectic structure which depends on the modular weight of the connected component of the critical set in which the Liouville torus is lying and the modular weights of the associated toric action. This action-angle theorem allows us to identify a neighborhood of the Liouville torus with the $b^m$-cotangent lift of the action of a torus acting by translations on itself.

    We end up this thesis proving a KAM theorem for $b^m$-Poisson manifolds which clearly refines and improves the one obtained for $b$-Poisson manifolds in \cite{KMS16}. As an outcome of this result together with the extension of the desingularization techniques of Guillemin-Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem for folded symplectic manifolds where KAM theory has never been considered before. In the way, we also obtain a brand new KAM theorem for symplectic manifolds where the perturbation keeps track of a distinguished hypersurface. In celestial mechanics this distinguished hypersurface can be the line at infinity or the collision set.