arXiv research

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

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

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

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

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 define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of $SL(2,{\mathbb R})$. We prove a fuchsian affine action of a surface group is never proper.

New insights into surface group actions and entropy.

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j,…

The paper characterizes geometrically finite surfaces via geodesic covers.

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

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

Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1, 1). In such a manifold, the minimal surface is unique and embedded, hence one ca…

The paper studies surface quotients of Fuchsian buildings.

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…

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

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

Smooth convergence to equidistant surfaces in Fuchsian manifolds.

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

Study shows limits of Fuchsian surfaces in hyperbolic 3-manifolds.

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

The paper finds representations of surface groups in SO(4,1) with specific curvature properties.

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.

New method finds Fuchsian representations dominating others in surface group representations.

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

Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.

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…

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…

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

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

An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the minimal surface is unique and embedded, hence one can parametrize these three-manifolds by…

Paper constructs Lawson surfaces using Fuchsian DPW potentials.

Proofs for spectral and geometric properties of hyperbolic surfaces.

New representations preserve hyperbolicity but not Fuchsian property.

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 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 $M$.

Characterizes monodromies of projective structures on finite-type surfaces.

In this paper, for a non compact and orientable surface $S$ been either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we construct explicitly an infinitely generated Fuchsian group $Γ<PSL(2,\mathbb{R})$, such that the quotient $\mathbb{H}/Γ$ is a hyperbolic surface homeomorphic to $S$.

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…

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…

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

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.

The study describes maximal Fuchsian subgroups of a specific Bianchi group and computes their covolumes.

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…

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

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…

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.

The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.