We establish a number of foundational results on Poincaré spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there exists a finite CW pair satisfying relative Poincaré duality in dimension …
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48 results for “Poincaré's result”
New tool: relative Hopf invariant for Poincaré surgery.
problem Non-simply connected Poincaré surgery.
method Relative Hopf invariant in equivariant setting.
result Established Poincaré embedding results in relative setting.
The paper shows plentiful non-homotopy finite Poincaré duality spaces.
problem The existence of non-homotopy finite Poincaré duality spaces.
method Constructing a finitely dominated Poincaré space with a non-trivial 2-divisible element in the reduced Grothendieck group.
result The existence of finitely dominated Poincaré spaces that are not homotopy finite.
The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.
problem Improving Poincaré and log-Sobolev inequalities on hyperbolic spaces.
method Establishing scale-dependent Poincaré-Hardy type identities and choosing suitable parameters, potentials, and vector fields.
result Derives new versions and substantially improves existing inequalities.
Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.
problem Understanding the structure of complete manifolds with specific curvature and inequality conditions.
method Analyzing manifolds with weighted Poincaré inequality and Ricci curvature bounds.
result Obtained splitting results for manifolds with non-zero weight function limit at infinity.
The singular sets of degenerate and nonlocal elliptic equations on Poincaré-Einstein manifoldsmath.DG
The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
problem Understanding the singular sets of degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
method Developing quantitative differentiation theory, stratification, Minkowski estimates, and ε-regularity results.
result Uniform Hausdorff measure estimates for the singular sets of degenerate/singular elliptic operators.
Researchers describe and compare decompositions of Poincaré duality pairs.
problem Understanding and comparing different decompositions of Poincaré duality pairs.
method Developed and described edge splittings of decompositions based on group properties.
result Compared decompositions with two other related decompositions.
The paper classifies Poincaré complexes as topological manifolds.
problem Classifying Poincaré complexes as topological manifolds.
method Using spherical fibrations and CW-complexes, the paper proves stability and homotopy equivalence.
result A sufficient condition for Poincaré complexes to be homotopy types of topological manifolds.
Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.
problem Rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.
method Established ε-regularity for Weyl curvature and proved rigidity results.
result Any Poincaré-Einstein filling of must be hyperbolic if non-positively curved.
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
Proves energy expression on Poincaré-Einstein spaces.
problem Computing renormalized Yang-Mills energy on Poincaré-Einstein manifolds.
method Generalizes Chang-Qing-Yang method for renormalized volumes and uses scattering theory for Schrödinger operators.
result Agrees with anomaly boundary integrand in seven dimensions.
The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincaré inequalities. Typically two kind of sensitivity indices are linked by a Poincaré type inequality, which provide upper bounds of the most interpretable index by using the other one, cheaper …
The Poincaré map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincaré map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of limit cycles (…
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant -Poincaré spaces and gap conditions.
result Space of isovariant structures on semifree -Poincaré spaces is highly connected.
On Poincaré Surgerymath.AT
Proves Poincaré surgery theorem using homotopy theory.
problem Fundamental Theorem of Poincaré surgery in simply connected spaces.
method Homotopy theoretic proof.
result Deduced Poincaré transversality exact sequence.
Develops parametrised Poincaré duality for equivariant fixed points.
problem Understanding equivariant fixed points in non-presentable settings.
method Introduces parametrised Poincaré duality in parametrised higher category theory, proving basechange results.
result Generalises Cnossen's twisted ambidexterity to non-presentable settings and applies to isotropy separation methods.
Extends Poincaré-Lefschetz duality to pairs of ∞-categories.
problem Generalizing Poincaré-Lefschetz duality to ∞-categories.
method Introduces Poincaré duality pairs of ∞-categories and uses them to study various diagrams of spaces.
result Unified treatment of Wall's Poincaré ads and iterated Poincaré cobordisms.
Uniform Poincaré inequalities established for various metric spaces.
problem Establishing uniform Poincaré inequalities on different metric spaces.
method Proper geodesic metric spaces equipped with a Borel measure. Local Poincaré inequality and volume conditions are used to derive uniform Poincaré inequalities.
result Uniform Poincaré inequalities are established for various metric spaces including hyperbolic spaces and covers of compact spaces.
The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.
problem Proving rigidity results for hypersurfaces under curvature constraints.
method Using a divergence formula for symmetric endomorphisms, the article deduces a Poincaré type inequality and applies it to higher-order mean curvature of hypersurfaces.
result The article proves several rigidity results for complete r-minimal hypersurfaces.
Mixing of the Mineyev flow, orbital counting and Poincaré series for strongly hyperbolic metricsmath.DS
Study on counting orbits and Poincaré series for specific hyperbolic metrics.
problem Counting orbits and analyzing Poincaré series for strongly hyperbolic metrics.
method Combining ergodic theory techniques with topological flows and symbolic dynamics.
result Obtained orbital counting results and described the domain of analyticity for Poincaré series.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
Explains the Borromean rings, icosahedron, and Poincaré homology sphere.
problem Exploring the relationship between Borromean rings, icosahedron, and Poincaré homology sphere.
method Introduction of topological concepts and geometric construction of icosahedral compound of octahedra.
result Proofs about the orientation-preserving symmetry group of an icosahedron and the linked nature of Borromean rings.
Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariantsmath.GT
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about t…
Short note proves Poincaré inequality for 4-manifold forms.
problem Quantifying Poincaré inequality for one forms on 4-manifolds.
method Hodge theory on orbifolds, comparison of fundamental groups, spectral convergence, degeneration to orbifolds.
result First non-trivial global Poincaré inequality without higher curvature assumptions.
The article proves Randers Poincaré disc satisfies isoperimetric equality.
problem Extending Riemannian isoperimetric equality to Finslerian case.
method Analyzes Randers Poincaré disc with different volume forms.
result Osserman's result cannot be extended to Finslerian case.
We state and prove a generalization of the Poincaré-Hopf index theorem for manifolds with boundary. We then apply this result to non-vanishing complex vector fields.
Enhanced loop space decomposition for specific Poincaré complexes.
problem Decomposing the loop space of certain high-dimensional complexes.
method Utilizing a result from BT2 to simplify and extend Beben and Wu's work.
result Improved understanding of the loop space structure of -connected -dimensional Poincaré Duality complexes.
A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…
The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.
problem Existence of Poincaré embeddings for specific spaces.
method Relates total obstruction to Reidemeister trace and uses Poincaré duality.
result Diagonal maps admit Poincaré embeddings under certain conditions.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
Analytic convex bodies' Poincaré series extended holomorphically.
problem Analytic continuation of Poincaré series for convex bodies.
method Analytic continuation of Laplace transforms, holomorphic functions, and resolvent of multiplication operators.
result Poincaré series continues holomorphically to a conical neighborhood of the right half-plane, removing countable cuts and points.
Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.
problem Proving non-vanishing of Poincaré series on finite-volume quotients of Hermitian symmetric spaces.
method Using Bergman kernels and averaging over discrete groups, proving non-vanishing of Poincaré series.
result Large class of relative Poincaré series does not vanish on general locally symmetric spaces of finite volume.
In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the cla…
New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.
problem Proving non-degeneracy of Poincaré-Einstein metrics.
method Proved non-degeneracy for 4D metrics satisfying a chiral curvature inequality.
result 4D Poincaré-Einstein metrics are non-degenerate if curvature is negative definite.
Study on filling 3D metrics with 4D Poincaré-Einstein structures.
problem Finding a conformal filling by a Poincaré-Einstein metric in 4D.
method Compactness result for conformally compact Einstein 4-manifolds under invariant conditions, with a rigidity result for hyperbolic metrics.
result Established compactness results and derived existence results for conformal fillings.
In this paper, we studied integrals involving both real and complex Hessian operators over bounded domain. Poincare type inequalities were proved in both cases which generalized a early results of Trudinger and Wang.
The Hirzebruch -genus and Poincare polynomial share some similar features. In this article we investigate two of their similar features simultaneously. Through this process we shall derive several new results as well as reprove and improve some known results.
Proves 3D Poincaré duality groups without property (T)
problem Residually finite 3D Poincaré duality groups and property (T)
method Using coboundary expansion and recent results on 3-manifold groups
result 3D Poincaré duality groups without property (T)
The paper characterizes Ricci solitons on the Poincaré upper half plane.
problem Characterizing Ricci solitons on the Poincaré upper half plane.
method Classifying and generalizing Ricci solitons and soliton equations in the half plane of Poincaré.
result Obtained some properties of solitons about their geodesic flows.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to varieties with isolated singularities.
method Using generalizations of the Poincaré-Hopf index.
result A Poincaré-Hopf type theorem for projective varieties with isolated singularities.
This paper extends the convergence analysis of Langevin Monte Carlo beyond Poincaré inequalities.
problem Analyzing convergence of Langevin Monte Carlo under various functional inequalities.
method Establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincaré inequalities.
result Explicitly quantifies the effect of the initializer on the performance of LMC algorithm.
New subsets without interior support Poincaré inequalities, expanding previous results.
problem Finding subsets without interior that satisfy Poincaré inequalities.
method Employing uniform domains and measure density, focusing on boundary regularity and separation.
result Existence of subsets supporting Poincaré inequalities without interior, applicable to various spaces.
We obtain multirelative connectivity statements about spaces of Poincare embeddings, as precursors to analogous statements about spaces of smooth embeddings. The latter are the key to convergence results in the functor calculus approach to spaces of embeddings.
A special knot in the Poincaré sphere leads to a unique connected sum of lens spaces.
problem Understanding the unique connected sum of lens spaces formed by a special knot in the Poincaré sphere.
method Analyzing the Seifert fibering and Dehn surgery of the Poincaré homology sphere.
result The only knot in the Poincaré sphere with a surgery to a connected sum of more than two lens spaces is the one mentioned.
We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green's function vanishing at infinity. On…
For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's result…
Classifies knots in the Poincaré sphere, using fixed points and folding automata.
problem Classifying knots in the Poincaré sphere and understanding their properties.
method Theory of train tracks, folding automata, and knot Floer homology.
result Almost completely classified genus-two, hyperbolic, fibered knots.