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G. P. Paternain

Publications and source records attributed to G. P. Paternain.

4 recordsLinked to original sources
arXiv2010

Symplectic topology of Mañé's critical values

We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mane critical value c. Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and it is invariant under under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k>c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is non-zero for energy levels k>c but vanishes for k<c, so levels above and below c cannot be connected by a stable tame homotopy. Moreover, we show that for strictly 1/4-pinched negative curvature and non-exact magnetic fields all sufficiently high energy levels are non-stable, provided that the dimension of the base manifold is even and different from two.

math.SG↗
arXiv2009

Stability is not open

We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.

math.SG↗
arXiv2007

The boundary rigidity problem in the presence of a magnetic field

For a compact Riemannian manifold with boundary, endowed with a magnetic potential $α$, we consider the problem of restoring the metric $g$ and the magnetic potential $α$ from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mañé action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of $α$. For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on $α$. We also show injectivity and stability for $g$ and $α$ in a generic class $\mathcal{G}$ including real analytic ones. For the nonlinear problem, we show rigidity for real analytic simple $g$, $α$. Also, rigidity holds for metrics in a given conformal class, and locally, near any $(g,α)\in \mathcal{G}$.

math.DG↗