My publications and preprints, with a short description of each:

We show the existence of several examples of geometric transition of four-dimensional real projective structures from a hyperbolic structure to an Anti-de Sitter structure through an half-pipe structure (as introduced by Danciger). The construction is rather explicit and relies on a four-dimensional polytope introduced by Kerckhoff and Storm, from which the four-manifolds are produced by certain glueing operations.

We study surfaces of constant affine curvature in three-dimensional equiaffine space, by obtaining a result of existence, and properness, of constant affine curvature surfaces with prescribed asymptotic behavior. This problem is strictly related to partial differential equations of Monge-Ampère type.

This work provides a classification of properly embedded surfaces of constant (or prescribed) Gaussian or constant mean curvature in three-dimensional Minkowski space. This is possible by a general existence result, which improves several previous results (including [4]) and whose main step is the use of a barrier constructed by the study of certain harmonic maps from the complex plane to an ideal triangle in the hyperbolic plane.

We generalize a well-known theorem of Wolpert in hyperbolic geometry, which gives an explicit formula for the Weil--Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics. We study here a more general object, namely a balanced geodesic graph, define an infinitesimal deformation associated to it, an prove an analogous of Wolpert's formula which reduces to his result when the geodesic graph is composed of disjoint simple closed curves.

Through the analytical study of two one-parameter families of examples, this paper discusses the optimal values of the constant C in the inequality log K ≤ C ||f||, which was proved in [10], where ||f|| is the cross-ratio norm of a quasisymmetric homeomorphism f and K is the maximal dilatation of its minimal Lagrangian extension to the hyperbolic plane.

This is the continuation of my first paper, together with my master’s advisor: this concerns the isometry group of spherical three-dimensional orbifolds, again using both algebraic and topological techniques. We prove that the natural homomorphism of π_0(Isom(O)) into π_0(Diff(O)) is a group isomorphism, for O a 3-dimensional spherical orbifold: this is the so-called π_0-part of the Smale Conjecture, for spherical 3-orbifolds.

I studied maximal surfaces in Anti-de Sitter three-dimensional space. By relating the curvature of a maximal surface to some other geometric invariants, such as the width of the convex hull, I obtained the following theorem, which is stated purely in terms of 2-dimensional conformal geometry: given any quasi-symmetric homeomorphism f of the circle, the quasiconformal dilation of the (unique) minimal Lagrangian extension F : D → D of f to the unit disc satisfies log K ≤ C ||f||, where C is a universal constant and ||f|| is the cross-ratio norm of f.

It is known that to every spacelike equivariant embedding of the universal cover of closed surface in Anti-de Sitter space, one can associate a Lagrangian surface in the product of two closed hyperbolic surfaces, for the standard symplectic structure. For instance, if the embedding has vanishing mean curvature, then the associated surface is minimal Lagrangian. We characterize the Lagrangian surfaces obtained in this way, by the condition of being Hamiltonian isotopic to the minimal Lagrangian surface.

This is a survey paper, which will appear on a book intended for a general mathematical audience. We describe several classical geometries from a projective point of view, and link them through the notion of geometric transition.

Using constant curvature surfaces in Anti-de Sitter space, we proved that every orientation-preserving homeomorphism of the circle admits a 1-parameter family (depending on a real parameter θ) of extensions F_θ : H^2 → H^2 which are θ-landslides: this means that F = f ◦ g^{-1} where f and g are harmonic maps whose Hopf differentials differ by multiplication by e^{iθ}.

This is a survey on the classification of spacelike surfaces of constant Gaussian curvature in Minkowski space of dimension (2+1).

This paper gives an alternative proof of the mail result of [9], thus proving the correspondence between Hamiltonian symplectomorphisms between closed hyperbolic surfaces and equivariant maps into Anti-de Sitter space. The proof of this paper uses mostly differential-geometric tools.

We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds M containing a closed orientable surface S, which (after Mess’ work) are parameterized by T(S)×T(S), similarly to quasi-Fuchsian manifolds. We prove that this volume is coarsely equivalent to the minima of the L^1-energy functional between the two hyperbolic surfaces which represent the two parameters in T(S).

We studied constant curvature surfaces in Minkowski space. The standard embedding of H^2 in Minkowski 3-space is not the unique isometric embedding of H^2; a classification is essentially given by the datum of a function f : S^1 → R, which encodes the asymptotic behavior. We proved the existence of constant curvature surfaces for any f lower-semicontinuous and bounded - thus improving the previously known results - and we proved that f has the Zygmund regularity if and only if the corresponding surface has bounded second fundamental form.

In this paper I studied minimal embedded disc in H^3. I proved that, if the boundary at infinity of a minimal disc S in H^3 is a K-quasicircle in ∂H^3, then the principal curvatures ±λ of S satisfy ||λ||∞ ≤ C log K, where C is a universal constant. This implies that the locus of almost-Fuchsian manifolds in T(S)×T(S) (i.e. in the deformation space of quasi-Fuchsian manifolds, T(S) being the Teichmüller space of a closed surface) contains a uniformly thick neighborhood of the diagonal, where “thickness” is measured by means of the Teichmüller distance.

This is the first paper of my PhD, and it concerns a differential-geometric approach to Mess' classification of maximal globally hyperbolic Minkowski 3-manifolds. Differently from Mess' techniques, our approach is suitable for generalizing to manifolds with conical singularities, and this enables us to prove a parameterization of the moduli space of such singular manifolds.

This paper originated from my master's thesis, and is written together with my master's advisor. It is about the classification of finite subgroups of SO(4) on the 3-sphere, and the understanding of the quotient orbifolds. The techniques of this paper are rather algebraic and topological, hence somewhat different from what I started doing in my PhD later on.

My PhD thesis, defended in December 2015, which also includes the results in [2, 3, 4, 11].