arXiv subjects
Publications and source records attributed to Roberto Giambó.
Using nonsmooth critical point theory, we prove the existence of at least N orthogonal geodesic chords in a class of Riemannian N-disk with strongly concave boundary. This yields a proof of a celebrated conjecture by H.Seifert on the number of brake orbits in a potential well of a natural Lagrangian/Hamiltonian system.
The complete spectrum of the endstates - naked singularities, or blackholes - of gravitational collapse is analyzed for a wide class of $N$-dimensional spacetimes in spherical symmetry, which includes and generalizes the dust solutions and the case of vanishing radial stresses. The final fate of the collapse is shown to be fully determined by the local behavior of a single scalar function and by the dimension $N$ of the spacetime. In particular, the ``critical'' behavior of the N=4 spacetimes, where a sort of phase transition from black hole to naked singularity can occur, is still present if N=5 but does not occur if $N > 5$, independently from the initial data of the collapse. Physically, the results turn out to be related to the kinematical properties of the considered solutions.
The dynamics of homogeneous Robertson--Walker cosmological models with a self-interacting scalar field source is examined here in full generality, requiring only the scalar field potential to be bounded from below and divergent when the field diverges. In this way we are able to give a unified treatment of all the already studied cases - such as positive potentials which exhibit asymptotically polynomial or exponential behaviors - together with its extension to a much wider set of physically sensible potentials. Since the set includes potentials with negative inferior bound, we are able to give, in particular, the analysis of the asymptotically anti De Sitter states for such cosmologies.
Conditions under which gravity coupled to self interacting scalar field determines singularity formation are found and discussed. It is shown that, under a suitable matching with an external space, the boundary, if collapses completely, may give rise to a naked singularity. Issues related to the strength of the singularity are discussed.
The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested in investigating the problem of existence of brake orbits and homoclinic orbits, in which case the Maupertuis--Jacobi principle produces a Riemannian manifold with boundary and with metric degenerating in a non trivial way on the boundary. In this paper we use the classical Maupertuis--Jacobi principle to show how to remove the degeneration of the metric on the boundary, and we prove in full generality how the brake orbit and the homoclinic orbit multiplicity problem can be reduced to the study of multiplicity of orthogonal geodesic chords in a manifold with {\em regular} and {\em strongly concave} boundary.