arXiv research

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

Alternative construction of quasi-Fuchsian flows using vortex equations.

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…

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…

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

We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.

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

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…

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…

Formula derived for FUP exponent in quasi-Fuchsian groups.

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

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…

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) …

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

The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.

In this paper, we will determine the topological types of hyperbolic 3-anifolds H^3/G such that G is a geometric limit of any algebraically convergent sequence of quasi-Fuchsian groups.

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

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.

This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.

The paper controls the geometry of surface subgroups in specific Kleinian groups.

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer $N>0$, we construct a quasi-Fuchsian manifold which contains at least $2^N$ such minimal surfaces. As a consequence, there exists so…

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…

We prove that a sequence of quasi-Fuchsian representations for which the critical exponent converges to the topological dimension of the boundary of the group (larger than 2), converges up to subsequence and conjugacy to a totally geodesic representation.

In this paper, we prove that if a quasi-Fuchsian 3-manifold $M$ contains a simple closed geodesic with complex length $\Lscr=l+iθ$ such that $θ/l\gg{}1$, then it contains at least two minimal surfaces which are incompressible in $M$.

Proofs for spectral and geometric properties of hyperbolic surfaces.

In this paper, we prove that if a quasi-Fuchsian 3-manifold contains a minimal surface whose principle curvature is less than 1, then it admits a foliation such that each leaf is a surface of constant mean curvature. The key method that we use here is volume preserving mean curvature flow.

Study mean curvature flow in hyperbolic 3-manifolds, proving foliations and existence of minimal surfaces.

We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with clo…

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…

Study on Cheeger constant in specific hyperbolic 3-manifolds.

For a given quasi-Fuchsian representation $ρ:π_1(S)\to$ PSL$_2\mathbb{C}$ of the fundamental group of a closed surface $S$ of genus $g\geq 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $ρ$ and two branch points is obtained by bubbling some unbranched structure on $S$ with the sa…

The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved t…

Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold $M$ is bounded by an explicit constant, depending only on the topology of $M$, times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of …

The paper shows nearly-Fuchsian properties for certain hyperbolic 3-manifolds.

Let $M$ be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function $H$ (with mild conditions) on $M$, we construct a closed incompressible surface with mean curvature $H$ . A direct application is the existe…

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…

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

Study of spectral properties of Lorentzian quasi-Fuchsian manifolds.

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…

The study bounds the number of quasi-Fuchsian surface subgroups in hyperbolic 3-manifolds.

We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective struc…

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_…

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…

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

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

We examine the dependence of the deformation obtained by bending quasi-Fuchsian structures on the bending lamination. We show that when we consider bending quasi-Fuchsian structures on a closed surface, the conditions obtained by Epstein and Marden to relate weak convergence of arbitrary laminations to the convergence …

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

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…