Fragility and Persistence of Leafwise Intersections

In this paper we study the question of fragility and robustness of leafwise intersections of coisotropic submanifolds. Namely, we construct a closed hypersurface and a sequence of Hamiltonians $C^0$-converging to zero such that the hypersurface and its images have no leafwise intersections, showing that some form of the contact type condition on the hypersurface is necessary in several persistence results. In connection with recent results in continuous symplectic topology, we also show that $C^0$-convergence of hypersurfaces, Hamiltonian diffeomorphic to each other, does not in general force $C^0$-convergence of the characteristic foliations.Comment: 17 pages, 3 figures; we removed one of our results (a refinement of Moser's theorem on leafwise intersections) and its proof, since a stronger theorem is proved in arXiv:1408.457

The implicit equation of a canal surface

A canal surface is an envelope of a one parameter family of spheres. In this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres. By using Laguerre and Lie geometries, we relate the equation of the canal surface to the equation of a dual variety of a certain curve in 5-dimensional projective space. We define the \mu-basis for arbitrary dimension and give a simple algorithm for its computation. This is then applied to the dual variety, which allows us to deduce the implicit equations of the the dual variety, the canal surface and any offset to the canal surface.Comment: 26 pages, to be published in Journal of Symbolic Computatio

Geometric Transitions

The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in mathematics and in physics.Comment: 44 page

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-dimensional Minkowski space for datas that are invariant under the action of a co-compact Fuchsian group