arXiv research

The study classifies weakly almost Fuchsian manifolds and proves geometric properties.

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…

Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma \to SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of Fuchsian representations, ie. composition of the inclusions of Gamma in SO(1,n) …

We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the cla…

A new metric model for quasi-Fuchsian space defined by Bers metrics.

The hyperbolic space $ \H^d$ can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $Γ$ is a group of linear isometries of the Minkowski space such that $\H^d/Γ$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, g…

We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a funct…

Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections…

New insights into surface group actions and entropy.

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuch…

We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely…

This paper shows how to approximate CAT(-1) representations by Fuchsian ones.

Study on Cheeger constant in specific hyperbolic 3-manifolds.

In this paper we give the characterization of Fuchsian groups acting on quaternionic hyperbolic 2-space.

The aim of this paper is to relate Thurston's metric on Teichmüller space to several ideas initiated by T. Sorvali on isomorphisms between Fuchsian groups. In particular, this will give a new formula for Thurston's asymmetric metric for surfaces with punctures. We also update some results of Sorvali on boundary isomorp…

Study compares metrics from negative curvature and quasi-Fuchsian representations.

The article constructs Fuchsian Schottky groups with conformal boundaries.

We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core.

In this paper, we study the topology of the boundaries of quasi-Fuchsian spaces. We first show for a given convergent sequence of quasi-Fuchsian groups, how we can know the end invariant of the limit group from the information on the behaviour of conformal structures at infinity of the groups. This result gives rise to…

The curvature of almost Fuchsian immersions is concave in their Hopf differentials.

We show a few propositions in favour of relations between the phase space of 3D gravity, moduli of quasi-Fuchsian groups, global solutions of cosh-Gordon equations and minimal surfaces in hyperbolic spaces.

Let $Γ$ be a finitely generated group, and let $\op{Rep}(Γ, \SO(2,n))$ be the moduli space of representations of $Γ$ into $\SO(2,n)$ ($n \geq 2$). An element $ρ: Γ\to \SO(2,n)$ of $\op{Rep}(Γ, \SO(2,n))$ is \textit{quasi-Fuchsian} if it is faithful, discrete, preserves an acausal subset in the conformal boundary $\Ein_…

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic…

Donaldon constructed a hyperkähler moduli space $\mathcal{M}$ associated to a closed oriented surface $Σ$ with $\textrm{genus}(Σ) \geq 2$. This embeds naturally into the cotangent bundle $T^*\mathcal{T}(Σ)$ of Teichmüller space or can be identified with the almost-Fuchsian moduli space associated to $Σ$. The later is t…

Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.

The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.

The paper studies curvature properties of vector bundles and their applications to quasi-Fuchsian space.

Paper studies complex Lagrangian surfaces and their relation to $\mathrm{SL}(3,\mathbb{C})$-representations.

The study shows how energy density of harmonic maps dominates in $n$-Fuchsian fibers, leading to unique minimal surfaces.

Paper constructs Lawson surfaces using Fuchsian DPW potentials.

We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projecti…

New framework mated Kleinian groups with complex polynomials, revealing unique group properties.

An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichmüller space is represented as a path $γ$, we show that $γ$ joins the …

We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy $ρ:π_1(S)\to$ PSL$_2\mathbb…

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.

Surveying Hitchin representations of Fuchsian groups.

We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. It follows that Mostow rigidity holds for Fuchsian buildings: if a group acts properly and cocompactly on tw…

Limit sets of $\mathrm{AdS}$-quasi-Fuchsian groups of $\mathrm{PO}(n,2)$ are always Lipschitz submanifolds. The aim of this article is to show that they are never $\mathcal{C}^1$, except for the case of Fuchsian groups. As a byproduct we show that $\mathrm{AdS}$-quasi-Fuchsian groups that are not Fuchsian are Zariski d…

Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.

Given a pants decomposition $\mathcal{PC} = \{γ_1, \ldots, γ_ξ\}$ on a hyperbolizable surface $Σ$ and a vector $\underline{c} = (c_1, \ldots, c_ξ) \in \mathbb{R}_+^ξ$, we describe a plumbing construction which endows $Σ$ with a complex projective structure for which the associated holonomy representation $ρ$ is quasi-F…

We prove that the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds is non-negative, with equality only for Fuchsian manifolds.

Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.

The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.

Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.

Study $\mathbb{C}$-Fuchsian subgroups of non-arithmetic lattices.

Alternative construction of quasi-Fuchsian flows using vortex equations.

This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there i…