arXiv subjects

Andrea Seppi

Publications and source records attributed to Andrea Seppi.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the diffeomorphism type of Seifert fibered spherical 3-orbifolds

It is well known that, among closed spherical Seifert three-manifolds, only lens spaces and prism manifolds admit several Seifert fibrations which are not equivalent up to diffeomorphism. Moreover the former admit infinitely many fibrations, and the latter exactly two. In this work, we analyse the non-uniqueness phenomenon for orbifold Seifert fibrations. For any closed spherical Seifert three-orbifold, we determine the number of its inequivalent fibrations. When these are in a finite number (in fact, at most three) we provide a complete list. In case of infinitely many fibrations, we describe instead an algorithmic procedure to determine whether two closed spherical Seifert orbifolds are diffeomorphic.

math.GT↗

arXiv2020

Affine deformations of quasi-divisible convex cones

For any subgroup of $\mathrm{SL}(3,\mathbb{R})\ltimes\mathbb{R}^3$ obtained by adding a translation part to a subgroup of $\mathrm{SL}(3,\mathbb{R})$ which is the fundamental group of a finite-volume convex projective surface, we first show that under a natural condition on the translation parts of parabolic elements, the affine action of the group on $\mathbb{R}^3$ has convex domains of discontinuity that are regular in a certain sense, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all these domains and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature. The proof is based on a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. As an independent result, we show that the moduli space of such groups is a vector bundle over the moduli space of finite-volume convex projective structures, with rank equaling the dimension of the Teichmüller space.

math.DG↗

arXiv2020

Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces

A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert's formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.

math.GT↗

arXiv2020

Anti-de Sitter geometry and Teichmüller theory

The aim of these notes is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichmüller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.

math.GT↗

arXiv2020

Isometry groups and mapping class groups of spherical 3-orbifolds

We study the isometry groups of compact spherical orientable $3$-orbifolds $S^3/G$, where $G$ is a finite subgroup of $\mathrm{SO}(4)$, by determining their isomorphism type. Moreover, we prove that the inclusion of $\mbox{Isom}(S^3/G)$ into $\mbox{Diff}(S^3/G)$ induces an isomorphism of the $π_0$ groups, thus proving the $π_0$-part of the natural generalization of the Smale Conjecture to spherical $3$-orbifolds.

math.GT↗

arXiv2020

Examples of geometric transition in low dimensions

The purpose of this note is to discuss examples of geometric transition from hyperbolic structures to half-pipe and Anti-de Sitter structures in dimensions two, three and four. As a warm-up, explicit examples of transition to Euclidean and spherical structures are presented. No new results appear here; nor an exhaustive treatment is aimed. On the other hand, details of some elementary computations are provided to explain certain techniques involved. This note, and in particular the last section, can also serve as an introduction to the ideas behind the four-dimensional construction of [RS19].

math.GT↗

arXiv2019

Complete CMC hypersurfaces in Minkowski (n+1)-space

We prove that any regular domain in Minkowski space is uniquely foliated by spacelike constant mean curvature (CMC) hypersurfaces. This completes the classification of entire spacelike CMC hypersurfaces in Minkowski space initiated by Choi and Treibergs. As an application, we prove that any entire surface of constant Gaussian curvature in 2+1 dimensions is isometric to a straight convex domain in the hyperbolic plane.

math.DG↗

arXiv2019

Spherical, hyperbolic and other projective geometries: convexity, duality, transitions

Since the end of the 19th century, and after the works of F. Klein and H. Poincaré, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen as a "limit" of both geometries. Then all the geometries that can be obtained in this way. Some of these geometries had a rich development, most remarkably hyperbolic geometry and the Lorentzian geometries of Minkowski, de Sitter and anti-de Sitter spaces, which in higher dimension have had large interest for a long time in mathematical physics and more precisely in General Relativity. Moreover, some degenerate spaces appear naturally in the picture, namely the co-Euclidean space (the space of hyperplanes of the Euclidean space), and the co-Minkowski space (that we will restrict to the space of space-like hyperplanes of Minkowski space), first because of duality reasons, and second because they appear as limits of degeneration of classical spaces. In fact, co-Minkowski space recently regained interest under the name half-pipe geometry. The purpose of the present paper is to provide a survey on the properties of these spaces, especially in dimensions 2 and 3, from the point of view of projective geometry. Even with this perspective, the paper does not aim to be an exhaustive treatment. Instead it is focused on the aspects which concern convex subsets and their duality, degeneration of geometries and some properties of surfaces in three-dimensional spaces. The presentation is intended to be elementary, hence containing no deep proofs of theorems, but trying to proceed by accessible observations and elementary proofs.

math.DG↗

arXiv2019

Regular domains and surfaces of constant Gaussian curvature in three-dimensional affine space

Generalizing the notion of domains of dependence in the Minkowski space, we define and study regular domains in the affine space with respect to a proper convex cone. In dimension three, we show that every proper regular domain is uniquely foliated by a particular kind of surfaces with constant affine Gaussian curvature. The result is based on the analysis of a Monge-Ampère equation with extended-real-valued lower semicontinuous boundary condition.

math.DG↗

arXiv2019

On the maximal dilatation of quasiconformal minimal Lagrangian extensions

Given a quasisymmetric homeomorphism $φ$ of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension $f_φ:\mathbb{H}^2\to\mathbb{H}^2$ to the hyperbolic plane. By previous work of the author, its maximal dilatation satisfies $\log K(f_φ)\leq C||φ||$, where $||φ||$ denotes the cross-ratio norm. We give constraints on the value of an optimal such constant $C$, and discuss possible lower inequalities, by studying two one-parameter families of minimal Lagrangian extensions in terms of maximal dilatation and cross-ratio norm.

math.CV↗

arXiv2018

Entire surfaces of constant curvature in Minkowski 3-space

This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a consequence of our classification of surfaces with bounded prescribed Gaussian curvature, sometimes called the Minkowski problem, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Some applications to minimal Lagrangian self-maps of the hyperbolic plane are obtained.

math.DG↗

arXiv2018

Area-preserving diffeomorphism of the hyperbolic plane and K-surfaces in Anti-de Sitter space

We prove that any weakly acausal curve $Γ$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$. The curve $Γ$ is the graph of a quasisymmetric homeomorphism of the circle if and only if the $K$-surfaces have bounded principal curvatures. Moreover in this case a uniqueness result holds. The proofs rely on a well-known correspondence between spacelike surfaces in Anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic plane. In fact, an important ingredient is a representation formula, which reconstructs a spacelike surface from the associated area-preserving diffeomorphism. Using this correspondence we then deduce that, for any fixed $θ\in(0,π)$, every quasisymmetric homeomorphism of the circle admits a unique extension which is a $θ$-landslide of the hyperbolic plane. These extensions are quasiconformal.

math.DG↗

arXiv2017

The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry

Given a smooth spacelike surface $Σ$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $ρ:π_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $Σ$ a diffeomorphism $ϕ_Σ$ of $S$. It turns out that $ϕ_Σ$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $ρ$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that $ϕ_Σ$ is the composition of a Hamiltonian symplectomorphism of $(S,h)$ and the unique minimal Lagrangian diffeomorphism from $(S,h)$ to $(S,h')$.

math.GT↗

arXiv2017

On the volume of Anti-de Sitter maximal globally hyperbolic three-manifolds

We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface $S$, in relation to some geometric invariants depending only on the two points in Teichmüller space of $S$ provided by Mess' parameterization - namely on two isotopy classes of hyperbolic metrics $h$ and $h'$ on $S$. The main result of the paper is that the volume coarsely behaves like the minima of the $L^1$-energy of maps from $(S,h)$ to $(S,h')$. The study of $L^p$-type energies had been suggested by Thurston, in contrast with the well-studied Lipschitz distance. A corollary of our result shows that the volume of maximal globally hyperbolic Anti-de Sitter manifolds is bounded from above by the exponential of (any of the two) Thurston's Lipschitz asymmetric distances, up to some explicit constants. Although there is no such bound from below, we provide examples in which this behavior is actually realized. We prove instead that the volume is bounded from below by the exponential of the Weil-Petersson distance. The proof of the main result uses more precise estimates on the behavior of the volume, which is proved to be coarsely equivalent to the length of the (left or right) measured geodesic lamination of earthquake from $(S,h)$ to $(S,h')$, and to the minima of the holomorphic 1-energy.

math.GT↗

arXiv2017

Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms

We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric homeomorphism $ϕ$, we study the relation between the width of the convex hull of the graph of $ϕ$, as a curve in the boundary of infinity of Anti-de Sitter space, and the cross-ratio norm of $ϕ$. As an application, we prove that if $ϕ$ is a quasisymmetric homeomorphism of $\mathbb{R}\mathrm{P}^1$ with cross-ratio norm $||ϕ||$, then $\ln K\leq C||ϕ||$, where $K$ is the maximal dilatation of the minimal Lagrangian extension of $ϕ$ to the hyperbolic plane.

math.DG↗

arXiv2017

Equivariant maps into Anti-de Sitter space and the symplectic geometry of $\mathbb H^2\times \mathbb H^2$

Given two Fuchsian representations $ρ_l$ and $ρ_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_ρ=(\mathbb{H}^2/ρ_l(π_1(S)))\times (\mathbb{H}^2/ρ_r(π_1(S)))$ and $ρ$-equivariant embeddings $σ$ of $\widetilde S$ into Anti-de Sitter space, where $ρ=(ρ_l,ρ_r)$ is the corresponding representation into $\mathrm{PSL}_2\mathbb R\times \mathrm{PSL}_2\mathbb R$. It is known that, if $σ$ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold $Λ_{\mathrm{ML}}$ of $M_ρ$. We show that, given any $ρ$-equivariant embedding $σ$, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to $Λ_{\mathrm{ML}}$. Conversely, any Lagrangian submanifold Hamiltonian isotopic to $Λ_{\mathrm{ML}}$ is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.

math.GT↗

arXiv2016

Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Ampère equation $\det D^2 u(z)=(1/ψ(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=φ$, for $ψ$ a smooth positive function and $φ$ a bounded lower semicontinuous function. We then prove that a domain of dependence $D$ contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function $φ$ is in the Zygmund class. Moreover in this case the surface of constant curvature $K$ contained in $D$ has bounded principal curvatures, for every $K<0$. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of $\partial \mathbb{D}$. Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature $K$, as $K$ varies in $(-\infty,0)$.

math.DG↗

arXiv2016

Minimal discs in hyperbolic space bounded by a quasicircle at infinity

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the boundary of a totally geodesic plane. As a by-product we prove that there is a universal constant C independent of the genus such that if the Teichmüller distance between the ends of a quasi-Fuchsian manifold $M$ is at most C, then $M$ is almost-Fuchsian. The main ingredients of the proofs are estimates on the convex hull of a minimal surface and Schauder-type estimates to control principal curvatures.

math.DG↗