Vittorio Martino

Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems

Abstract We prove integral formulas for closed hypersurfaces in ℂ n+1, which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the “Soap Bubble Theorem” for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.

A Legendre Transform on an Exotic S 3

Abstract We consider an exotic contact form α on S3 and we establish explicitly the existence of a non singular vector field v in ker(α) such that the non-singular one-differential form β(・) := dα(v, ・) is a contact form on S3 with the same orientation as α. In particular this means that a Legendre transform can be completed.

The topology of a subspace of the Legendrian curves on a closed contact 3-manifold

Abstract In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.

Legendre duality on hypersurfaces in Kähler manifolds

Abstract We give a sufficient condition on real strictly Levi-convex hypersurfaces M, embedded in four-dimensional Kähler manifolds V , such that Legendre duality can be performed. We consider the contact form θ on M whose kernel is the restriction of the holomorphic tangent space of V and show that if there exists a Legendrian Killing vector field ν, then the dual form β ( · ) := dθ(ν; · ) is a contact form on M with the same orientation than θ.

Some integral formulas for the characteristic curvature

ABSTRACT We show some integral formulas involving the characteristic curvature for closed real hypersurfaces in complex spaces.

Palais–Smale sequences for the fractional CR Yamabe functional and multiplicity results

In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behaviour of the Palais–Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely many solutions to the related equation.