Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.
problem Singularity of hitting measure for random walks on discrete subgroups.
method Algebraic and geometric convergence, hyperbolic Dehn filling.
result Proved singularity conjecture for certain measures on cocompact Fuchsian and Kleinian groups.
The hitting measure is singular and has dimension less than 1 for cocompact Fuchsian groups.
problem Analyzing the hitting measure and Hausdorff dimension for cocompact Fuchsian groups.
method Geometric and probabilistic analysis of random walks on cocompact Fuchsian groups.
result The hitting measure is singular with respect to Lebesgue measure and has a Hausdorff dimension strictly less than 1.
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…
Desingularizes Einstein metrics with A1 singularities in 4D.
problem Desingularizing Einstein metrics with specific singularities.
method Recursive procedure to desingularize Fuchsian singularities.
result Desingularizations of non degenerate Poincaré-Einstein metrics with A1 singularities remain non degenerate.
Characterizes monodromies of projective structures on finite-type surfaces.
problem Understanding monodromies of projective structures on finite-type surfaces.
method Geometrical/topological study of local conical projective structures.
result Any representation can be represented as the holonomy of a branched projective structure.
New proof of past stability for Kasner solutions in (3+1)-dimensional Einstein vacuum spacetime.
problem Stability of Kasner singularities in (3+1)-dimensional Einstein vacuum spacetime. method Developed (2+1) orthonormal-frame decomposition and symmetrization argument, applying Fuchsian techniques. result Perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete, and crushing at the Big Bang singularity.
Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are 2-tangle q-fractions.
Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.
problem Domination problem between non-Fuchsian representations of closed surface groups.
method Analysis of branched harmonic immersions and construction of anti-de Sitter 3-manifolds.
result Found that representations admitting branched harmonic immersions dominate other representations, and constructed large families of branched anti-de Sitter 3-manifolds.
Study of holomorphic correspondences combining entire maps and Fuchsian groups.
problem Understanding dynamics of entire maps and their interactions with Fuchsian groups.
method Systematic study of (∞:∞) holomorphic correspondences arising from conformal combinations of transcendental entire maps and Fuchsian groups. result The resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function with a simple pole.
Let S be a compact surface of genus >1, and g be a metric on S of constant curvature K\in\{-1,0,1\} with conical singularities of negative singular curvature. When K=1 we add the condition that the lengths of the contractible geodesics are >2π. We prove that there exists a convex polyhedral surface P in the Lorentzian …
We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than π: any first-order deformation changes either one of those angles or the conformal …
Study of ants' movement rules on a 6D space, revealing distribution structures and singular trajectories.
problem Understanding the movement patterns of ants in a 6D space.
method Analyzing mechanical system rules to derive distribution structures and singular trajectories.
result Distributions and singular trajectories of ants' movement rules in a 6D space.
Study of defects in gauge theories connects quantum field theory to classical integrability.
problem Vacuum expectation values of half-BPS surface defects in gauge theories.
method Analysis of Fuchsian systems, isomonodromic deformations, and blowup formulas.
result Establishes a relation between supersymmetric gauge theory and classical integrability.
This paper's theme is the relation between several classical and well-known objects: triangle Fuchsian groups, quasi-homogeneous singularities of plane curves, torus knot complements in the 3-sphere. Torus knots are the only nontrivial knots whose complements admit transitive Lie group actions. In fact S^3\K_{p,q} is d…
The paper studies complex affine structures near irregular singularities.
problem Understanding complex affine structures near irregular singularities.
method Introducing local invariants and a Delaunay decomposition.
result Upper bounds on the complexity of the Delaunay decomposition.
The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.
problem Existence of monotone CMC foliations for quasi-Fuchsian manifolds.
method Analyzes quasi-Fuchsian manifolds near the Fuchsian locus and proves the existence of a unique monotone CMC foliation.
result Proves the existence of a unique monotone CMC foliation for quasi-Fuchsian manifolds in a small neighborhood of the Fuchsian locus.
We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface M=Γ\H2 associated with a Fuchsian group of the 1st kind Γ containing parabolic elements. M is t…
Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds…
Article shows AdS-quasi-Fuchsian groups' limit sets are not smooth.
problem Smoothness of limit sets of AdS quasi-Fuchsian groups.
method Analyzes limit sets of AdS quasi-Fuchsian groups in PO(n,2).
result Limit sets are never C^1, except for Fuchsian groups.
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…
New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.
problem Determining foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus.
method Inspired by Bonahon's method, uses measured bending laminations on the boundary of convex cores.
result Measured foliations at infinity of quasi-Fuchsian manifolds can be uniquely realized for small t. Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
problem Understanding foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
method Using measured foliations and quasi-Fuchsian manifolds, proving the existence and uniqueness of foliations by constant mean curvature surfaces.
result For quasi-Fuchsian manifolds close to the Fuchsian locus, measured foliations at infinity can be uniquely realized and foliated by constant mean curvature surfaces.
The study classifies weakly almost Fuchsian manifolds and proves geometric properties.
problem Classifying and understanding weakly almost Fuchsian manifolds.
method Geometric analysis and compactification techniques.
result Uniform upper bounds on volume and Hausdorff dimension for limit sets.
Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
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…
Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.
problem Understanding foliations at the boundary of quasi-Fuchsian manifolds.
method Proving filling pairs and showing realisation of measured foliations.
result Proved that measured foliations at infinity of quasi-Fuchsian manifolds are filling when close to being Fuchsian.
Study Liouville action on quasi-Fuchsian groups, proving formulas and relating to holography.
problem Analyzing Liouville action for quasi-Fuchsian groups with different types of elements.
method Derived formulas for classical Liouville action, proved first and second variations, and established holography principle.
result Established an equality linking Liouville action and renormalized volume for quasi-Fuchsian groups.
Solves Deligne-Simpson problem for special connections on Gm.
problem Existence of Fuchsian connections with specific singularities.
method Theory of fundamental and regular strata, lattice chain filtration, quiver varieties.
result Characterization of rigid connections with unipotent monodromy at infinity.
We show that the real-valued function Sα on the moduli space M0,n of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic 2-sphere with n≥3 conical singularities of arbitrary orders α={α1,...,αn}, generates accessory parameters of the as…
Study Fuchsian loci in mPSLn(R)-Hitchin components of a pair of pants.
problem Understanding Fuchsian loci in mPSLn(R)-Hitchin components. method Using Bonahon-Dreyer parametrization, explicit parametrization of Fuchsian loci of a pair of pants.
result Explicit parametrization of Fuchsian loci of a pair of pants.
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…
Locally rigid groups from 5-polytopes with Fuchsian ends.
problem Constructing locally rigid right-angled Coxeter groups with Fuchsian ends.
method Constructing a right-angled 5-polytope P and analyzing its right-angled Coxeter groups.
result Locally rigid right-angled Coxeter groups with Fuchsian ends can be constructed.
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.
problem Identifying compact Fuchsian manifolds with convex boundaries.
method Proving uniqueness based on the induced path metric on the boundary.
result Compact Fuchsian manifolds with convex boundaries are uniquely determined by the induced path metric on the boundary.
We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, H3, and on invariant disks embedded in H3. 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…
Computes cohomology groups for NEC groups, focusing on Fuchsian groups.
problem Understanding the cohomology of non-Euclidean crystallographic groups.
method Computes cohomology groups for geometrically finite NEC groups, and determines the ring structure for Fuchsian groups.
result Determination of cohomology groups and ring structures for Fuchsian groups.
The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.
problem Confirming a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.
method Proving the long-time existence and convergence of a modified mean curvature flow.
result The CMC foliation conjecture is confirmed for a subclass of almost Fuchsian manifolds.
Study on Cheeger constant in specific hyperbolic 3-manifolds.
problem Analyzing Cheeger constant in convex co-compact hyperbolic 3-manifolds.
method Examining Cheeger constant as a functional in the space of specific hyperbolic 3-manifolds.
result Global maximum of Cheeger constant is uniquely attained at the Fuchsian locus.
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) …
New findings on Fuchsian groups and their axes.
problem Determining the commensurability class of Fuchsian groups.
method Analyzing geodesic intersections and using a new approach.
result The conjecture holds for almost all Fuchsian groups.
Paper connects Painlevé VI equation to irregular systems, solving monodromy data.
problem Solving monodromy data for irregular systems related to Painlevé VI.
method Expressed Frobenius integrability in terms of PVI, computed monodromy data for coalescing eigenvalues.
result Computed monodromy data for transcendentals holomorphic at critical points of PVI.
Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.
problem Understanding the dimensionality of harmonic measures for random walks.
method Analyzing finite range random walks on Fuchsian Schottky groups.
result Harmonic measures have dimension strictly less than the limit set's Hausdorff dimension.
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. Alternative construction of quasi-Fuchsian flows using vortex equations.
problem Constructing quasi-Fuchsian flows.
method Using coupled vortex equations to construct quasi-Fuchsian flows as thermostats.
result Formulas for the marked length spectrum of quasi-Fuchsian flows.
Totally geodesic limit of quasi-Fuchsian groups with converging critical exponent.
problem Understanding convergence of quasi-Fuchsian groups.
method Critical exponent convergence analysis.
result Convergence of quasi-Fuchsian groups to totally geodesic representations.
The paper shows nearly-Fuchsian properties for certain hyperbolic 3-manifolds.
problem Characterizing hyperbolic 3-manifolds with specific surface properties.
method Analyzing minimal and non-minimal surfaces with principal curvatures in [-1,1](-1,1).
result Weakly almost-Fuchsian manifolds are nearly-Fuchsian.
We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesaro averages of spherical averages in a Fuchsian group.
The trace set of a Fuchsian group Γ ist the set of length of closed geodesics in the surface Γ\H. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithm…