Geometria diferencial: foliacions lagrangianes, mètriques singulars i integrals de corvatura en espais hiperbòlics

This project deals with several aspects of differential geometry and has four research lines corresponding to the following objectives: a) Local and semilocal simplectic classification of Langragian foliations associated to non-degenerate Hamiltonian systems completely integrable, study of its equivariant versions. b) Classification of degenerations of metrics with cone singularities (or angle pi) in three-dimensional manifolds. Analysis of possible degenerations for larger cone angles, and topological applications. c) Study of the asymptotic behaviour of the quotients of curvature integrals for sequences of convex sets expanding over the whole hyperbolic space. This will generalize some previous results on the quotient area/volume previously considered by our research group d) Determine the possible bounds for the total curvature integral of compact manifolds immersed in the hyperbolic space. Generalization to hyperbolic space of the theory of tight immersions