3-manifolds with positive scalar curvature and bounded geometry are contractible.
problem Characterizing 3-manifolds with positive scalar curvature and bounded geometry.
method Maximal weak solution to inverse mean curvature flow.
result Complete contractible 3-manifolds with positive scalar curvature and bounded geometry are R3. Study of hyperbolic 3-manifolds via fractional Dehn twists and cusp geometry.
problem Understanding the geometry of fibred hyperbolic 3-manifolds via combinatorial data.
method Relating Euclidean cusp geometry to fractional Dehn twist coefficients of monodromies.
result Uniform bounds on fractional Dehn twist coefficients for certain open book decompositions.
Study shows genus two Heegaard diagrams for all Thurston geometries.
problem Finding Heegaard diagrams for Thurston geometries.
method Collecting genus two Heegaard diagrams for all eight Thurston geometries.
result Each Thurston geometry has a genus two Heegaard example.
New geometric structures on 3-manifolds discovered and proven for all closed orientable ones.
problem Existence of specific geometric structures on 3-manifolds.
method Generalized geometry and surgery techniques.
result Any closed orientable 3-manifold admits a B3-generalized complex structure. 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.
3-manifolds with specific homology are cobordant if homeomorphic.
problem Characterizing integer homology cobordism for Sol manifolds.
method Analyzing first homology groups and using cobordism theory.
result Integer homology cobordant Sol manifolds are homeomorphic.
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.
Study curves of constant breadth in a specific 3D manifold.
problem Differential geometry of curves in Walker 3-manifolds.
method Investigate curves of constant breadth using Darboux frame.
result Properties of curves of constant breadth in Walker 3-manifolds.
Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.
problem Understanding the geometry of fully augmented links in doubled 3-manifolds.
method Constructing fully augmented links on the reflection surface of doubled 3-manifolds and finding bounds on cusp shapes and volumes.
result Bounds on cusp shapes and volumes of hyperbolic links in doubled 3-manifolds.
Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.
problem Understanding the structure of diffeomorphism groups of 3-manifolds.
method Deformation retraction and earlier work by multiple authors combined.
result Homotopy type of Diff(X) determined for prime 3-manifolds.
3-manifold groups have a property that allows them to act on quasi-trees.
problem Proving property (QT) for 3-manifold groups.
method Analyzing the geometry of 3-manifolds and using quasi-trees.
result Compact, orientable 3-manifold groups have property (QT) if they don't support Sol or Nil geometry.
Neumann's work connects 3-manifold invariants to quantum topology.
problem Understanding invariants of 3-manifolds.
method Ideal triangulations, volume, Chern-Simons invariants, Bloch group.
result Neumann's work bridges classical invariants to quantum topology.
New polynomial connects knot genus to 3-manifold geometry.
problem Detecting knot genus from 3-manifold geometry.
method Ideal triangulation and super-Ptolemy assignments.
result New polynomial conjecturally agrees with torsion polynomial.
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
New method for computing hyperbolic structures on 3-manifolds with torus boundaries.
problem Computing a complete hyperbolic structure on 3-manifolds with torus boundaries.
method Convex optimization and combinatorial modifications to find a triangulation that admits a solution to the gluing equations.
result Experimental results support the new method for modifying triangulations and updating their geometry.
3D hyperbolic spaces have endless simple paths.
problem Finding simple paths in complex 3D spaces.
method Analyzing geodesics in hyperbolic 3-manifolds.
result Cusped hyperbolic 3-manifolds have infinitely many simple closed geodesics.
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.
Improved bounds linking entropy and volume in hyperbolic 3-manifolds.
problem Establishing bounds between entropy and volume in hyperbolic 3-manifolds.
method Heegaard Floer homology and hyperbolic geometry.
result Entropy is bounded by hyperbolic volume with logarithmic factor.
Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
We define some new invariants for 3-manifolds using the space of taut codim-1 foliations along with various techniques from noncommutative geometry. These invariants originate from our attempt to generalise Topological Quantum Field Theories in the Noncommutative geometry / topology realm.
We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involvin…
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.
Article explores Thurston's circle packing theorem in 3-manifold geometry.
problem Understanding Thurston's circle packing theorem in 3-manifold geometry.
method Analyzes the Koebe-Andre'ev-Thurston Theorem and its relation to Thurston's circle packing theorem.
result Illustrates the significance of Thurston's circle packing theorem in 3-manifold geometry.
Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.
problem Proving certain hyperbolic link complements don't contain geodesic 3-manifolds.
method Analyzing hyperbolic link complements of 2-tori in S^4.
result Proves certain hyperbolic link complements do not contain closed embedded totally geodesic hyperbolic 3-manifolds.
The paper finds hyperbolic small knots in many 3-manifolds.
problem Finding small knots in 3-manifolds.
method Explicit examples of hyperbolic small knots in spherical 3-manifolds.
result Explicit examples of hyperbolic small knots in most spherical 3-manifolds.
Study connects surface projections in fibered 3-manifolds.
problem Understanding surface projections in fibered 3-manifolds.
method Uniformly relates structure of surface projections across fibrations.
result Extends previous work to general cases.
3-manifolds can be virtually dominated by maps of degree 8.
problem Understanding virtual domination of 3-manifolds.
method Proving existence of finite covers with specific map properties.
result Virtual 8-dominance of 3-manifolds.
The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.
problem Checking cosmetic surgeries on knots and 3-manifolds.
method Knot invariants and hyperbolic geometry.
result Verification of no cosmetic surgeries on knots and 3-manifolds.
In this paper we determined all of the possible self mapping degrees of the manifolds with S3-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self mapping degrees of all closed orientable 3-manifold in Thurston's picture.
3-manifolds with similar completions have matching slopes and polynomials.
problem Matching slopes and polynomials in cusped hyperbolic 3-manifolds.
method Prove similarity of profinite completions leads to matching A-polynomials and boundary slopes. result Strongly detected boundary slopes match between manifolds with similar completions.
Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.
problem Characterize the Lipschitz homotopy groups of contact 3-manifolds.
method Sub-Riemannian geometry, geometric measure theory, biLipschitz equivalence, purely unrectifiable sets.
result Contact 3-manifolds are K(π,1) spaces with uncountably generated first homotopy groups. Improved pathwidth bound for hyperbolic 3-manifolds.
problem Finding a more efficient triangulation for hyperbolic 3-manifolds.
method Synthesis of tools from 3-manifold theory: Heegaard splittings, amalgamations, and thick-thin decomposition.
result Volume provides a linear upper bound on pathwidth of triangulations.
The paper examines timelike conformal fields on 3-manifolds and finds they are rigidly tied to specific geometric structures.
problem Investigating timelike conformal vector fields on closed Lorentzian 3-manifolds.
method Performing conformal changes to unit vectors and analyzing the resulting flows through stable Hamiltonian structures and cohomology.
result Timelike conformal vector fields on 3-manifolds are either Reeb vector fields of Sasakian or co-Kähler structures.
Study G2-manifolds from symplectic SU(3)-manifolds with T2-symmetry.
problem Understanding G2-manifolds from symplectic SU(3)-manifolds. method Cohomological lifting of multi-toric graphs.
result Compact part of G2-moment graph can be obtained cohomologically from the base. 3-manifold triangulations are Golod and tight, proven through a topological characterization.
problem Understanding Golodness and tightness in 3-manifold triangulations.
method Topological characterization of a polyhedral product for a tight-neighborly manifold triangulation.
result Golodness and tightness are equivalent for 3-manifold triangulations.
We study the large-scale geometry of 3-manifolds with nontrivial 2-dimensional bounded cohomology, with a view to proving a weak version of the geometrization conjecture for such manifolds.
New Skyrme model for contact geometry with topological solutions.
problem Finding BPS solutions for maps between contact 3-manifolds.
method Defined a new Skyrme energy functional for maps between contact 3-manifolds and showed existence of solutions to a first-order self-duality equation.
result Existence of solutions to the Beltrami maps equation, generalizing the original Ferreira-Zakrzewski model.
Study examines corrections to heterotic geometry on SU(3) manifolds.
problem Analyzing α′2 corrections to heterotic supersymmetry algebra. method Derives integrability condition and pure gauge correction from graviton equation of motion.
result Curvature of tangent bundle connection acquires a (0,2) component, disrupting semi-classical intuition.
We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize whi…
Link between Teichmüller and anti de Sitter geometry via length functions.
problem Understanding the geometry of Teichmüller space and anti de Sitter manifolds.
method Establishing a connection between Teichmüller space and anti de Sitter geometry through length functions.
result New purely anti de Sitter proofs of Teichmüller theory results.
Study on fibered knots in 3-manifolds, proving unrelated volume and genus.
problem Unrelated volume and genus in hyperbolic fibered knots.
method Proof of unrelated volume and genus for hyperbolic fibered knots in 3-manifolds.
result Proved that volume and genus are unrelated for hyperbolic fibered knots.
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
Study of harmonic maps to the circle with applications to hyperbolic 3-manifolds.
problem Understanding harmonic maps and their properties.
method Analyzing maps from 3-manifolds to the circle, proving dihedral rigidity.
result Special case of dihedral rigidity of three dimensional cubes with π/2 angles. No CMC surfaces exist in certain hyperbolic 3-manifolds.
problem Existence of constant mean curvature (CMC) surfaces in hyperbolic 3-manifolds.
method Proof by contradiction using hyperbolic geometry and topology.
result Nonexistence of CMC surfaces with mean curvature ≥ 1.
We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M < 3.44, then H_1(M;Z/2Z)…
The study connects surface geometry in 5D to 4D projections and umbilic curvatures.
problem Understanding the geometry of surfaces in 5D space.
method Relating surfaces in 5D to surfaces in 4D via projections and normal sections, analyzing asymptotic directions and umbilic curvatures.
result Relations between asymptotic directions and umbilic curvatures in 5D surfaces and their counterparts in 4D projections.
Book introduces hyperbolic geometry for knot theory.
problem Understanding knots through hyperbolic geometry.
method Explains hyperbolic geometry, geometric structures, and techniques.
result Develops three knot invariants from hyperbolic geometry.
In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating combinatorial descriptions and geometric properties of hyperbolic 3-manifolds.