Pubblicazioni

Bifurcations on contact manifolds J. Fixed Point Theory and Applications. 119 - 138. Vol. 20 (3). (Springer Nature) (2018)

Consider a 1-parameter compactly supported family of Legendrian submanifolds of the 1-jet bundle of a compact manifold with its natural contact structure and a path of intersection points of the Legendrian family with the 1-jet of a constant function. Since the contact distribution is a symplectic vector bundle, it is possible to assign a Maslov type index to the intersection path. We show that the non-vanishing of the Maslov intersection index implies that there exists at least one point of bifurcation from the given path of intersection points. This result can be viewed as a kind of analogue in bifurcation theory of the Arnold-Sandon conjecture on intersections of Legendrian submanifols. The proof is based on the technique of generating functions, that relates the properties of Hamiltonian diffeomorphisms to the Morse theory of the associated functions.
Bifurcations on the 1-jet bundle. Proceedings of the Third EAUMP Conference. (2016) Makerere University, Kampala, Uganda. 166-173.
Bifurcation results on symplectic manifolds.. Proceedings of the Second EAUMP Conference. (2012) The Nelson Mandela Institute of Science and Technology. Arusha, Tanzania. 108-119.
A bifurcation theorem for lagrangian intersections (with J. Pejsachowicz). Progress on Nonlinear Differential Equations and Their Applications. 105 - 115. Vol. 40 editato da Jürgen Appell. (Birkhauser) (2000)

The main result is as follows. Let N be a closed manifold and let L = {L_t} be an exact, compactly supported family of Lagrangian submanifolds of the symplectic manifold M = T_*(N) such that L₀ admits a generating family quadratic at infinity. Let p:[0,1]® M be a path of intersection points of L_t with N. Assume that L_t is transversal to N at p(t) for t = 0,1 and that the Maslov intersection index m(L,N,P) is different from zero. Then arbitrarily close to the branch p there are intersection points of L_t with N such that do not belong to p.

Lower bounds for contraction constants of non-zero degree mappings onto the sphere. (with M. Llarull). Diff. Geom. and its applications 14 (2001), n.2, 209-216.

This paper studies contraction constants of non-zero degree mappings from compact spin Riemannian manifolds onto the standard Riemannian sphere. Assuming uniform lower bound for the scalar curvature, we find a sharp lower bound for the dilation constants in terms of the dimension of the sphere. In the best case, we prove rigidity.

Bifurcation of periodic orbits of time dependent Hamiltonian systems on symplectic manifolds .Rendiconti del Seminario Matematico dell'Universita e del Politecnico di Torino. Vol. 57 n.3 (1999) 161-173.

For a 1-parameter family of time dependent Hamiltonian vector fields, acting on a symplectic manifold M which possesses a known trivial branch u_l of 1-periodic solutions it is shown that if the relative Conley Zehnder index of the monodromy path along u_l (0) is defined and does not vanish then any neighborhood of the trivial branch contains 1-periodic solutions not in the branch. This result is applied to bifurcation of fixed points of Hamiltonian symplectomorphisms when the first Betty number of M vanishes.

Uniqueness of spectral flow. (with M. Fitzpatrick and J. Pejsachowicz). Journal of mathematical modeling and computation. 32 (Birkhauser) (2000) n. 11-13, 1495-1501.

On special submanifolds in Symplectic Geometry. Differential Geometry and its applications 3 (1993) 91-99.

McDuff proved that the Kähler form w on a simply connected complete Kähler 2n-dimensional manifold P of non-positive curvature is diffeomorphic to the standard symplectic form w₀ on R²ⁿ. We show that the symplectomorphism she constructed takes totally geodesic complex (therefore symplectic) submanifold Q into complex (therefore symplectic) linear subspace of R²ⁿ. She also proved that if L is a properly embeded totally geodesic Lagrangian submanifold of (P,w) then P is symplectomorphic to the cotangent bundle T^*L with its usual symplectic structure. We extend this result to the case of totally geodesic isotropic submanifolds of P.

The local structure of a Liouville vector field. American Journal of Mathematics 115 n. 4 (1993) 735-747.

` In this work we investigate the local structure of a Liouville vector field x of a Kähler manifold (P,W) which vanishes on an isotropic submanifold Q of P. Some of the eigenvalues of its linear part at the singular points are zero and we assume that the remaining ones are in resonance. We show that for any positive integer K there is a C^K-smooth linearizing conjugation between the Liouville vector field xand its linear part. To do this we construct Darboux coordinates adapted to the unstable foliation which is provided by the Center Manifold Theorem. We then apply linearization result due to G. Sell.

Symplectomorphic codimension 1 totally geodesic submanifolds. Differential Geometry and its applications 5 (1995) 99-104.

Here we continue the study of special submanifolds of (P,w). We show that also the coisotropic totally geodesic properly embeded submanifolds of codimension 1 are linearizable. First we show that such a submanifold is foliated by totally geodesic complex leaves transversal to an isometric flow hence by a result of E. Ghys is a Riemannian product. We then apply a result of B. Reinhart.

Symplectic Geometry of Morse singularity. Seminari de Geometria 1991-1993. Universitaà di Bologna. Edited by S.Coen. 81-87.

In this notes we discuss the symplectic properties of Morse singularity

f(z₁,...,z_n) = z₁²+...+z_n+1²: Cⁿ⁺¹® C.

We verify that the diffeomorphism between the non-singular fiber M_c = f^-1(c)È[`B]₂, 0 < |c| < 2 with the unit disc subbundle of the cotangent bundle of Sⁿ induces a symplectic structure on M_c. We identify a Lagrangian representative of the generator of the homology group Hⁿ(M_c) and of the relative homology group H_n(M_c,¶M_c). We also show that the boundary of the last Lagrangian submanifold is a Legendrian submanifold of the contact manifold ¶M_c.