The paper explores the formality of low-dimensional manifolds using algebraic structures.
problem Investigating the formality of low-dimensional manifolds.
method Introducing Poincaré DGCAs of Hodge type and small algebra/quotient algebras to study equivalence classes of manifolds.
result A (r−1) connected Poincaré DGCA of Hodge type is A∞-quasi-isomorphic to an A3-algebra, with the formality determined by a specific Harrison cohomology class. 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.
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected. 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.
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 paper proves new results on Poincaré duality pairs and spaces.
problem Establishing Poincaré duality in various contexts and dimensions.
method Analyzes Poincaré spaces and CW pairs, proving relative Poincaré duality and related results.
result Found a finite CW pair (X,Y) where Y fails to satisfy Poincaré duality in any dimension. Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
New extremal Poincaré type metrics found via blow-up theorem.
problem Finding new extremal Poincaré type metrics.
method Proved Arezzo-Pacard-Singer blow-up theorem for Poincaré type metrics and applied to new examples.
result Found new examples of extremal Poincaré type metrics with an additional obstruction.
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.
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.
Uniform rectifiability proven for sets with Poincaré inequalities.
problem Uniform rectifiability of sets with Poincaré inequalities.
method Weak (1,d)-Poincaré inequality and surface measure. result Uniform rectifiability achieved for sets supporting such inequalities.
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.
Study shows knots in Poincaré sphere can't be 'cosmetically' altered.
problem Cosmetic surgeries on knots in the Poincaré homology sphere.
method Proof that essential knots cannot undergo purely cosmetic surgeries.
result Essential knots in the Poincaré sphere cannot be 'cosmetically' altered.
Investigates conditions for Poincaré map existence and uniqueness in systems with impulse effects.
problem Existence and uniqueness of Poincaré maps for systems with impulse effects.
method Investigates sufficient conditions for the existence and uniqueness of Poincaré maps for dynamical systems with impulse effects evolving on a differentiable manifold.
result Shows sufficient conditions for the existence and uniqueness of Poincaré maps for systems with impulse effects.
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.
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.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to projective varieties with isolated singularities.
method Using generalized Poincaré-Hopf indices for a projective variety with isolated determinantal singularities.
result A Poincaré-Hopf type theorem is proven for projective varieties with isolated singularities.
Expands Euler-Poincare characteristic to supergeometry.
problem No new problem introduced.
method Introduced Euler-Poincare characteristic pair in supergeometry.
result Examined transversality in Π-symmetric supermanifolds. 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.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
Proves a theorem for 3D Poincaré duality pairs.
problem No specific problem stated; focuses on proving a theorem.
method Analogous to Johannson's theorem for PD3 pairs.
result Proves a theorem for 3D Poincaré duality pairs.
Optimal Poincaré constant estimates on manifolds with ends.
problem Estimating the Poincaré constant on manifolds with ends.
method Heat kernel estimates extended to manifolds with ends, focusing on central balls.
result The Poincaré constant is determined by the second largest end.
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.
We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.
The Poincaré series for surfaces with boundary extends to the complex plane.
problem Counting geodesics on surfaces with boundaries.
method Analytic continuation of Poincaré series.
result Poincaré series extend meromorphically to the whole complex plane.
The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.
problem The Nielsen realisation problem for aspherical manifolds.
method Application of equivariant Poincaré duality to cyclic groups of prime order.
result Removal of a technical condition in the Nielsen realisation problem.
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 S1imesSn−1 must be hyperbolic if non-positively curved. The paper studies Fox pairings of Poincaré duality groups using group cohomology.
problem Understanding Fox pairings of Poincaré duality groups.
method Using group cohomology, the paper computes cohomology groups of Fox pairings.
result The paper suggests fundamental and higher Fox pairings.
We prove two kinds of fibering theorems for maps X --> P, where X and P are Poincare spaces. The special case of P = S^1 yields a Poincare duality analogue of the fibering theorem of Browder and Levine.
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 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.
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
In this paper, using exclusively homotopy theoretical methods, we study degrees of maps between (n−2)-connected (2n−1)-dimensional Poincar\' e complexes which have torsion free integral homology. Necessary and sufficient algebraic conditions for the existence of map degrees between such Poincar\' e complexes are es…
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 study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
problem Proving rigidity for Poincaré-Einstein manifolds with specific conformal infinity.
method Analyzing curvature tensors over level sets of a boundary defining function.
result Rigidity theorem for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
We describe classes of potential structures (covector fields) on Minkowski space that admit subgroups of the Poincaré group. We describe also seven classes of Maxwell spaces that admit subgroups of the Poincaré group.
Sharp Poincaré inequality proved for specific metric spaces.
problem Proving a sharp Poincaré inequality for certain metric measure spaces.
method Identifying model densities and using localization arguments, without assuming geodesic convexity.
result Best possible Poincaré constant as a function of parameters.
Reduces field theories using Poisson-Poincaré method.
problem Reduction of field theories using Poisson-Poincaré method.
method Poisson-Poincaré reduction for field theories.
result Reduction procedure for field theories.
Study of Poincaré-Reeb graphs for algebraic domains.
problem Characterizing geometric shapes of algebraic domains.
method Collapsing vertical segments to form Poincaré-Reeb graphs and analyzing their properties.
result Any transversal graph with specific properties can be realized as a Poincaré-Reeb graph.
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.
Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.
Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.
problem Proving uniqueness of Poincaré type cscK metric with singularity at a smooth divisor.
method Holomorphic transformations, asymptotic behavior analysis, fixed point problem.
result Unique Poincaré type cscK metric with singularity at a smooth divisor is unique up to holomorphic transformations.
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with dis…
Characterizes solvability of J-equation on Kähler surfaces with singularities.
problem Solvability of J-equation on Kähler surfaces with Poincaré type singularities.
method Two-parameter continuity path for J-equation, Kähler metrics with Poincaré type singularities.
result Existence of Poincaré type solutions implies boundedness of K-energy.
Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.
problem Isometry and eigenvalue gap in Finslerian models.
method Presented models and isometry proofs.
result First eigenvalue gapless for all three models.
The paper constructs Poincaré-Einstein 4-manifolds with various cusps.
problem Constructing Poincaré-Einstein 4-manifolds with cusps.
method Constructing metrics on (0,∞)imesN and (0,∞)imesP. result Infinite families of Einstein metrics on (0,∞)imesN and (0,∞)imesP.