3-manifolds explained through geometry, proving Thurston's conjecture.
problem Thurston's conjecture about 3-manifolds.
method Explain geometrisation theorem and its proof by Perelman.
result Proved Thurston's conjecture about 3-manifolds.
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17 results for “Geometrisation”
Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an inco…
The paper geometrizes N-manifolds using symmetric vector bundles.
problem Geometrizing N-manifolds with a specific symmetry.
method Equivalence between -manifolds and symmetric -fold vector bundles.
result Identical cocycles between symmetric vector bundles and -manifolds.
New algorithm for recognizing 3-spheres using Groebner basis methods.
problem Recognizing 3-spheres among integer homology 3-manifolds.
method Groebner basis methods applied to SL(2,C) representations of fundamental groups.
result Existence of a new algorithm for 3-sphere recognition.
Study of knotoids using double branched covers and geometric tools.
problem Characterizing and distinguishing knotoids and knots.
method Using double branched covers and geometric invariants.
result Established a 1-1 correspondence between knotoids and knots with strong inversion.
This paper equates geometric structures to algebraic ones for degree 2 manifolds.
problem Equivalence of geometric structures to algebraic ones for degree 2 manifolds.
method Using involutive double vector bundles and linear metrics, the paper establishes an equivalence between canonical categories.
result Equivalence of Poisson involutive double vector bundles to Poisson N-manifolds of degree 2.
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.
Study of rod packings in 3-torus using 3-manifold geometry.
problem Understanding crystal structures in crystallography through rod packings in 3-torus.
method Use of 3-manifold geometry and topology to analyze complements of rod packings.
result Find families of complements that are hyperbolic and Seifert fibred.
Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metr…
In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly emp…
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.
Examines properties of negatively curved 3-manifolds and their embeddings, introduces Cross Curvature Flow.
problem Understanding negatively curved three-manifolds and their embeddings.
method Reviews Cross Curvature Flow as a tool, examines rigidity properties, reviews embeddings into Minkowski space.
result Fixed Einstein volume solutions are integrable solutions, answering a question posed by Chow and Hamilton.
Novel Lie-superalgebraic description of superstring dynamics.
problem Describing superstring dynamics in super-Minkowskian background.
method Cartan-Eilenberg super-1-gerbe geometrising higher gauge field.
result Manifestly supersymmetric model of supermanifolds.
We examine how generalised geometries can be associated with a labelled Dynkin diagram built around a gravity line. We present a series of new generalised geometries based on the groups for which the generalised tangent space transforms in a spinor representation of the group. In …
Scheme resolves super-brane topology via equivariant structures.
problem Topological issues in super-brane dynamics.
method Geometric scheme for Green-Schwarz cocycles, equivariant structures.
result Equivariant structures resolve super-brane topology.
Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has in…
Explores new geometric structures and their physics applications.
problem Exploring new geometric structures and their physics applications.
method Investigation of exotic differentiable structures and their relationship to physics.
result Development and exploration of new geometric structures.