Surfaces in constant curvature three-manifolds and the infinitesimal Teichmüller theory - TEL - Thèses en ligne Accéder directement au contenu
Thèse Année : 2015

Surfaces in constant curvature three-manifolds and the infinitesimal Teichmüller theory

Surfaces dans 3-variétés de courbure constante et la théorie infinitésimale de Teichmüller

Résumé

In this thesis are exploited several instances of the relationship between convex Cauchy surfaces S in flat Lorentzian (2+1)-dimensional maximal globally hyperbolic manifolds M and the tangent bundle of Teichmüller space $\mathcal{T}(S)$ of the topological surface S. This relationship was first pointed out by Geoffrey Mess in the case of closed surfaces. The first case presented is the case of simply connected surfaces, and M is a domain of dependence in $\mathbb{R}^{2,1}$. We prove a classification of entire surfaces of constant curvature in $\mathbb{R}^{2,1}$ in terms of Zygmund functions on the circle, which represent tangent vectors of universal Teichmüller space $\mathcal{T}(\D)$ at the identity. An important ingredient is the solvability of Minkowski problem for Cauchy surfaces in any domain of dependence M contained in the future cone over some point of $\mathbb{R}^{2,1}$, which is proved by analyzing the Dirichlet problem for the Monge-Amp\`ere equation $\det D^2 u(z)=(1/\psi(z))(1-|z|^2)^{-2}$ on the disc, where $\psi$ is a smooth positive function. Moreover, when S is a surface of constant curvature, the principal curvatures are bounded if and only if $\varphi$ is in the Zygmund class. The situation of S a closed surface, and M is a maximal globally hyperbolic flat spacetime diffeomorphic to $S\times \mathbb{R}$, is next discussed. We provide an explicit relation between the embedding data of any strictly convex Cauchy surface in M and the holonomy of M, which was used by Mess to parametrize the moduli space of manifolds M as above by means of the tangent bundle of $\mathcal{T}(S)$. The techniques used in this thesis are amenable to be extended to the case of globally hyperbolic flat spacetimes with $n>0$ particles, namely cone singularities along timelike lines, where the cone angle is assumed in $(0,2\pi)$. The analogue of Mess' parametrization is then proved, showing that the corresponding moduli space is parametrized by the tangent bundle of Teichmüller space of the closed surface S with n punctures. The above connections can be regarded as an infinitesimal version of the relation of Teichmüller space $\mathcal{T}(S)$ and universal Teichmüller space $\mathcal{T}(\D)$ with surfaces in maximal globally hyperbolic Anti-de Sitter manifolds (either with the topological type of a closed surface, or with trivial topology) and in quasi-Fuchsian hyperbolic manifolds (or in $\mathbb{H}^3$ itself). In the last part of the thesis this perspective is discussed, and the behavior of zero mean curvature surfaces in $\mathbb{H}^3$ and $AdS^3$ close to the Fuchsian locus is discussed. The main result in hyperbolic space is a sublinear estimate of the supremum of principal curvatures of a minimal embedded disc in $\mathbb{H}^3$ spanning a quasicircle $\Gamma$ in the boundary at infinity in terms of the norm of $\Gamma$ in the sense of universal Teichmüller space, provided $\Gamma$ is sufficiently close to being the boundary of a totally geodesic plane. As a by-product, there is a universal constant C such that if the Teichmüller distance between the ends of a quasi-Fuchsian manifold M is at most C, then M is almost-Fuchsian, independently of the genus. In Anti-de Sitter space, an estimate is proved for the principal curvatures of any maximal surface with boundary at infinity the graph of a quasisymmetric homeomorphism $\phi$ of the circle. The supremum of the principal curvatures is estimated again in a sublinear way, in terms of the cross-ratio norm of $\phi$. This also provides a bound on the maximal distortion of the quasiconformal minimal Lagrangian extension to the disc of a given quasisymmetric homeomorphism.
Dans ma thèse doctorale, j'ai étudié principalement les plongements de surfaces dans des 3-variétés Riemanniennes et Lorentziennes de courbure constante. J'ai écrit deux articles avec mon directeur de thèse Francesco Bonsante sur le cas des variétés Lorentziennes plates, l'un sur les surfaces convexes dans les espace-temps plats maximaux globalement hyperboliques, également quand on permet l'existence de singularités de type temps, et l'autre sur les surfaces convexes dans l'espace de Minkowski (qui est l'analogue Lorentzien de l'espace Euclidien) en relation avec la courbure Gaussienne. D'autre part, pendant mon séjour à l'Université du Luxembourg, j'ai commencé l'étude des surfaces à courbure moyenne nulle dans 3-variétés à courbure sectionnelle constante et négative. J’ai écrit un article qui concerne les surfaces minimales dans l'espace hyperbolique et un autre article concernent des surfaces maximales dans l'espace Anti-de Sitter, qui peut être considéré comme l'analogue Lorentzien de l'espace hyperbolique. Dans plusieurs cas, il y a une forte relation entre les surfaces plongées et la théorie de Teichmüller, en particulier la théorie des applications entre surfaces.
Fichier principal
Vignette du fichier
thesis.pdf (1.4 Mo) Télécharger le fichier
Loading...

Dates et versions

tel-01294224 , version 1 (10-04-2016)

Identifiants

  • HAL Id : tel-01294224 , version 1

Citer

Andrea Seppi. Surfaces in constant curvature three-manifolds and the infinitesimal Teichmüller theory. Mathematics [math]. Université de Pavie, Département de Mathématiques et Statistiques (Italie), 2015. English. ⟨NNT : ⟩. ⟨tel-01294224⟩

Collections

INSMI
100 Consultations
258 Téléchargements

Partager

Gmail Facebook X LinkedIn More