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…
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48 results for “geometrization conjecture”
In this paper we propose counterexamples to the Geometrization Conjecture and the Elliptization Conjecture.
Proves conjecture for hyperbolic-by-cyclic groups using geometric methods.
problem Proving Farrell-Jones Conjecture for specific group structures.
method Geometric methods and structure theory of mapping tori.
result Proved Farrell-Jones Conjecture for mapping tori of automorphisms of hyperbolic-by-cyclic groups.
Singer Conjecture for Varieties with Semismall Albanese Map and Residually Finite Fundamental Groupmath.DG
Proves Singer conjecture for specific geometric varieties.
problem Singer conjecture for certain geometric varieties.
method Semismall Albanese map and residually finite fundamental group.
result Proves Singer conjecture for specified varieties.
Reformulated Markov's conjecture in combinatorial terms.
problem Markov's uniqueness conjecture in integral necklaces.
method Geometric reformulation and combinatorial description.
result Explicitly described set of lengths on modular torus.
First geometric proof of the flyping theorem.
problem Proving Tait's flyping conjecture.
method Geometric proof using Greene's characterization, Menasco's crossing ball structures, and isotopy/re-plumbing moves.
result First entirely geometric proof of Menasco-Thistlethwaite's flyping theorem.
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.
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.
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.
New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.
problem Finding complete sub-Riemannian structures satisfying the Minimizing Sard conjecture.
method Techniques from nonsmooth analysis and geometric measure theory.
result Complete sub-Riemannian structures associated with distributions of co-rank 2 or generic distributions of rank ≥ 2 satisfy the Minimizing Sard conjecture.
The paper proposes conjectures about moduli space rigidity.
problem Rigidity of moduli spaces in algebraic geometry.
method Group-theoretic, topological, and holomorphic approaches.
result Some conjectures have been proved.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compres…
In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré con…
We prove the Farrell-Jones conjecture for free-by-cyclic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture.
Study supports conjecture about pretzel links' homology.
problem Investigate extreme Khovanov homology of pretzel links.
method Geometric realizations of the independence simplicial complex.
result Supports conjecture that extreme Khovanov homology is torsion-free.
Study geometric properties of generalized vacuum static spaces.
problem Estimating geometric properties of generalized -vacuum static spaces.
method Proving estimates for -scalar curvature and first eigenvalue of the Jacobi operator, and rigidity under various geometric assumptions.
result Proved a result related to the Cosmic no-hair conjecture.
This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. …
This paper has been withdrawn by the author due a crucial sign error in Theorem B. We present a geometric proof of Thom conjecture, which uses Khovanov homology. Our approach doesn't use any analytic methods and is quite different from proof given by Kronheimer and Mrowka in 1994.
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.
New geometric proof shows index of umbilic points on analytic surfaces is at most one.
problem Proving the Carathéodory Conjecture for compact simply connected embedded surfaces.
method Geometric analysis of degenerate umbilic points on analytic surfaces.
result Index of an umbilic on an analytic surface cannot be an integer larger than one.
Let be an orthogonal matrix with no entries zero. Let be the matrix defined by . M. Kontsevich conjectured that the rank of is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statment can be understood as a generalization of …
Disproves Fedotov's conjecture on higher-order Shephard inequalities.
problem Fedotov's conjecture on higher-order Shephard inequalities.
method Using Hodge-Riemann relations for simple convex polytopes.
result Fedotov's conjecture is disproved.
The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompr…
CAT(0) spaces of higher rankmath.MG
Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.
problem CAT(0) spaces of higher rank with geometric group actions.
method Proving rigidity for spaces containing periodic flats and geodesics in flats.
result CAT(0) spaces of higher rank are rigid if they contain a periodic -flat.
The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.
problem Proving the geometric stability conjecture for scalar curvature on manifolds conformal to tori.
method Reduction via the Yamabe problem and handling sequences of manifolds conformal to flat tori or constant negative scalar curvature.
result Proves the geometric stability conjecture for certain sequences of manifolds conformal to flat tori.
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.
New research disproves a key conjecture in optimization.
problem Comparison of sampling methods in stochastic optimization.
method Reduction to noncommutative arithmetic-geometric mean inequality and application of noncommutative Positivstellensatz.
result The Recht-Ré conjecture is false for general n.
Study explores unstable 3-forms on Calabi-Yau 3-folds.
problem Understanding degenerations of Calabi-Yau 3-folds via 3-forms.
method Investigates geometries of 3-forms on symplectic 6-manifolds.
result Unstable 3-forms reveal rich geometric properties related to SYZ conjecture.
We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates thi…
Geometric proof confirms link volume conjecture.
problem Volume conjecture for hyperbolic 3-manifolds.
method Topological and skein theoretic techniques.
result Complements of octahedral fully augmented links are isometric to fundamental shadow links.
Study proves volume conjecture for specific 3-manifolds.
problem Proving the Andersen-Kashaev volume conjecture for FAMED triangulations.
method Introducing FAMED triangulations and proving existence of Jones function.
result Proves the Andersen-Kashaev volume conjecture for FAMED geometric triangulations.
We give a survey of geometric approaches to the topological 4-dimensional surgery and 5-dimensional s-cobordism conjectures, with a focus on the study of surfaces in 4-manifolds. The geometric lemma underlying these conjectures is a statement about smooth immersions of disks and of certain 2-complexes, capped gropes, i…
The paper proves a quantum modularity conjecture for 3-manifolds.
problem Quantum invariants of 3-manifolds at roots of unity.
method Formulates and proves a strong version of the conjecture for geometric 3-manifolds.
result The conjecture holds for Brieskorn homology spheres and some other examples.
Geometric zeta functions of Ihara and Hashimoto are generalized to higher rank. The -adic version of the Patterson conjecture is proven.
The paper proves geometrical finiteness for automorphism groups of K3 surfaces and related varieties.
problem Establishing geometrical finiteness for automorphism groups of K3 surfaces and related varieties.
method Using cone conjecture, the paper establishes geometrical finiteness for the natural isometric actions of automorphism groups on hyperbolic spaces.
result Automorphism groups of K3 surfaces and related varieties are non-positively curved and relatively hyperbolic.
This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to …
The paper confirms conjectures about Stein manifolds formed by quotients of the ball.
problem Characterizing Stein manifolds formed by quotients of the ball.
method Analyzing discrete subgroups of PU(n,1) and their quotients.
result The quotient of the ball by geometrically finite groups is Stein.
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential in…
This is the announcement of an alternative approach to the 3-dimensional Poincaré Conjecture, different from Perelman's big and spectacular breakthrough. No claim concerning the other parts of the Thurston Geometrization Conjecture, come with our purely 4-dimensional line of argument.
The Kähler-Ricci flow smooths positive currents on Kähler manifolds.
problem Geometric regularization of positive closed currents on Kähler manifolds.
method Kähler-Ricci flow on compact Kähler manifolds.
result Local Arnold multiplicities linearly decrease to zero under the flow.
We recall fundamental aspects of the pluriclosed flow equation and survey various existence and convergence results, and the various analytic techniques used to establish them. Building on this, we formulate a precise conjectural description of the long time behavior of the flow on complex surfaces. This suggests an at…
The paper proves inequalities for closed surfaces involving mean curvature.
problem Proving geometric inequalities for closed surfaces in Euclidean space.
method Verification of inequalities for convex surfaces and addressing Topping's conjecture.
result Optimal scaling law between Willmore energy and isoperimetric ratio for convex surfaces.
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…
Surveying conjectures on compactness and scalar curvature.
problem Understanding sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces.
method Analyzing progress and related examples towards conjectures.
result Survey of current progress and related examples.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…