Explains Thurston geometries and visualization techniques.
problem Classifying Thurston geometries and their visualization.
method Survey of history and recent techniques.
result Discussion of recent immersive visualization techniques.
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48 results for “Thurston's geometrization”
This book explains Thurston's geometrization of 3-manifolds.
problem Understanding the geometry and topology of 3-manifolds.
method Detailed explanations of geometric and topological concepts.
result Thurston's geometrization theorem proved by Perelman.
Study of Thurston's Master Teapot and its properties.
problem Characterize geometric and topological properties of Thurston's Master Teapot.
method Establish basic geometric and topological properties through analysis of self-maps and intersections.
result The Master Teapot is connected, contains the unit cylinder, and its intersection with grows monotonically with .
This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
Historical notes on Thurston's 3-manifold geometry.
problem Exploring Thurston's notes on 3-manifold geometry.
method Historical commentary and references.
result Discussion of Lobachevsky, Andreev, and Milnor's works.
The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.
problem Determining if integral points on the Thurston norm dual ball correspond to geometric structures.
method Examining various geometric, topological, and dynamical structures on 3-manifolds.
result Integral points on the Thurston norm dual ball correspond to the Euler class of taut foliations and other structures.
Geometric properties of Anosov representations into symplectic group.
problem Geometric properties of Anosov representations.
method Geometric properties of Anosov representations into symplectic group.
result Geometric properties of Anosov representations into symplectic group.
Limits of limit sets Imath.MG
We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…
Linking manifold structures to group dynamics.
problem Understanding geometric structures on manifolds and group dynamics.
method Study of geometric structures and representations of discrete groups into Lie groups.
result Recent findings on connections between manifold structures and group dynamics.
The paper solves a geometric P=W conjecture for SL(2,C) using Thurston's compactification.
problem Addressing the geometric P=W conjecture for SL(2,C) in projective compactifications of character varieties of closed surfaces.
method Using Thurston's compactification of Teichmüller space and new results, the paper constructs a projective compactification of the SL(2,C)-character variety of any closed surface of genus g>1.
result The boundary divisors are toric varieties and the dual intersection complex is a sphere.
Solves a 45-year-old Poincaré Conjecture using geometrization of 3-manifolds.
problem Proving the Poincaré Conjecture for 3-manifolds.
method Combining four geometries in Lie form and using affine geometry for edge regions.
result Proves the Poincaré Conjecture for 3-manifolds using Thurston's geometrization.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…
Cannon-Thurston Mapsmath.GT
Cannon-Thurston maps link complex and hyperbolic studies of Kleinian groups.
problem Understanding connections between Kleinian groups and hyperbolic geometry.
method Overview and sketch of connections to hyperbolic subgroups and open questions.
result Theory of Cannon-Thurston maps links complex and hyperbolic studies of Kleinian groups.
Survey of combination theorems in geometry and dynamics.
problem Combination theorems in hyperbolic geometry, group theory, and dynamics.
method Survey and focus on Thurston's contributions.
result Thurston's influence on combination theorems.
Heegaard Floer homology bounds Thurston norm for certain knots and links.
problem Bounding the Thurston norm for specific knots and links.
method Using twisted correction terms in Heegaard Floer homology.
result Lower bounds on Thurston norm for strong concordance classes of 2-component links and knots.
Geometrization in Geometrymath.DG
Geometrization Theorem solves complex geometry problems.
problem Complex geometry problems in differential geometry.
method Based on Hamilton's program, proved by Grigory Perelman.
result Generalized Poincaré's Conjecture.
The paper shows Thurston geometries don't support Anosov diffeomorphisms.
problem The existence of Anosov diffeomorphisms on Thurston geometries.
method Analyzing Thurston geometries and their properties.
result Thurston geometries do not support transitive Anosov diffeomorphisms.
Study continuity of limit sets in symmetric spaces.
problem Continuity of limit sets for geometrically finite subgroups in symmetric spaces.
method Extended geometrically finite representations theory.
result Limit sets vary continuously with respect to Hausdorff distance under strong convergence.
The paper shows how profinite completions can reveal 4D geometries except for specific cases.
problem Detecting 4D geometries via profinite completions of fundamental groups.
method Analyzing profinite completions of fundamental groups of 4-manifolds.
result Not all 4D manifolds are geometric, but some Seifert fibred ones are.
Study geometric manifolds in arbitrary dimensions, focusing on maps and diffeomorphisms.
problem Existence and properties of maps and diffeomorphisms in geometric manifolds.
method Analysis of geometric structures and homotopy invariants in arbitrary dimensions.
result Existence of Anosov diffeomorphisms and monotonicity of homotopy invariants.
3D spaces collapse to 8 Thurston geometries.
problem Characterizing collapsed 3D Alexandrov spaces.
method Proving Alexandrov spaces modeled on Thurston geometries.
result Sufficiently collapsed irreducible Alexandrov 3-spaces are geometric.
The seven non euclidean geometries of the Thurston's geometrization program are proved to originate naturally from singularization morphisms and versal deformations on euclidean 3-manifolds generated in the frame of the Langlands global program. The Poincare conjecture for a 3-manifold appears as a particular case of t…
4-manifolds with non-hyperbolic geometry are ordered.
problem Ordering 4-manifolds with non-hyperbolic Thurston geometry.
method Derived from closed aspherical 4-manifolds and maps of non-zero degree.
result Kodaira dimension is monotone with maps of non-zero degree.
The paper describes geometric properties of Teichmüller space metrics.
problem Analyzing the geometry of Teichmüller space with weak Finsler metrics.
method Geometric description of unit spheres in weak Finsler metrics.
result Introduced a family of weak Finsler metrics interpolating between Thurston's metric and Teichmüller metric.
The paper proves inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.
problem Inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.
method Analyzes geometric -norms, Thurston norms, and Lipschitz maps to prove inequalities.
result Proves an inequality between geometric -norm and Thurston norm, qualitatively sharp.
Study geometric structures on LVM threefolds, focusing on resonant structures.
problem Understanding deformations of geometric structures on LVM threefolds.
method Using the Ehresmann-Thurston principle and Kuranishi family construction.
result Construction of a family containing all LVM threefolds and complete at every point.
Study of Teichmüller space geometry using infinitesimal and global methods.
problem Understanding the geometry of Teichmüller space and its tangent/cotangent spheres.
method Systematic study of Thurston metric's infinitesimal and global properties.
result Rigidity statements for the Thurston metric analogous to Royden theorem.
We study canonical decompositions of postcritically finite branched coverings of the 2-sphere, as defined by K. Pilgrim. We show that every hyperbolic cycle in the decomposition does not have a Thurston obstruction. It is thus Thurston equivalent to a rational map.
Random veering triangulations are not geometric in most cases.
problem Characterizing geometric properties of veering triangulations.
method Combining Teichmüller theory, Ending Lamination Theorem, Thurston norm study, and rigorous computation.
result Generically, veering triangulations are not geometric.
Study calculates -Alexander torsion for specific geometric spaces.
problem Calculating -Alexander torsion for specific geometric spaces.
method Using Thurston norm to express the -Alexander torsion for Seifert fiber spaces and graph manifolds.
result Successfully calculated -Alexander torsion for Seifert fiber spaces and graph manifolds.
This paper determines the flexible exponent for non-geometric 3-manifolds.
problem Bounding the mapping degree in terms of the Lipschitz constant for non-geometric 3-manifolds.
method Analyzing the infimum of α such that the inequality holds for any Lipschitz map.
result The flexible exponent for non-geometric 3-manifolds is determined.
We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows tha…
New discrete Ricci flow method resolves 3D geometrization.
problem Discretizing and solving Thurston's geometrization for 3D axially symmetric geometries.
method Discrete Ricci flow (DRF) with surgery, sparsely coupled edge length equations, Forman-Ricci tensor diagonalization.
result Explicit numerical realization of Thurston's geometrization for 3D axially symmetric neckpinch geometry.
Higgs bundles help describe geometric structures on manifolds.
problem Describing geometric structures on manifolds explicitly.
method Using Higgs bundles to construct geometric structures.
result Higgs bundles can be used to construct geometric structures in some cases.
The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.
problem Characterizing and understanding geometric properties of 4D projective manifolds.
method Analyzing geometric decompositions and using properties of locally symmetric spaces.
result Closed, indecomposable 4D projective manifolds are either real hyperbolic or have real hyperbolic pieces.
The Cannon-Thurston map's measures become singular with respect to sphere measures.
problem Characterizing the behavior of measures under the Cannon-Thurston map.
method Analyzing the geometric properties of hyperbolic geodesics and quasi-geodesics.
result Natural measures on the circle become singular with respect to measures on the sphere.
Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical sys…
Algorithm determines Thurston equivalence of topological polynomials.
problem Determining Thurston equivalence of topological polynomials.
method Geometric group theory, iteration on simplicial complex of trees.
result Algorithm produces Hubbard tree or canonical obstruction.
We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…
The paper classifies hypersurfaces in a specific 4D geometry.
problem Classify homogeneous hypersurfaces in the four-dimensional Thurston geometry .
method Used geometric conditions to classify hypersurfaces with constant principal curvatures.
result Complete classification of homogeneous hypersurfaces in .
In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery on geometrically finite hypebolic cone-manifolds.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M…
We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian group…
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
New metrics derived from Hölder distortion on Hitchin components.
problem Deriving metrics on Hitchin components from Hölder distortion.
method Expressing Thurston's metric in terms of Hölder regularity of boundary maps, associating stratified loci, and measuring relative Hölder distortion.
result First known geometrically significant complete metrics on Hitchin components for .
Geometric conemanifold structures on , the result of surgery in the left-handed trefoil knot math.GT
As an example of the transitions between some of the eight geometries of Thurston, investigated before, we study the geometries supported by the cone-manifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phe…
Surjectivity of Cannon-Thurston map proven for metric graph bundles.
problem Proving surjectivity of Cannon-Thurston map in metric graph bundles.
method Generalized Mj-Sardar's result to include more types of fibers.
result Continuous extension map between boundaries is surjective.