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.
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48 results for “Poincaré-invariant”
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.
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
The study shows that certain conformal classes on S^7 cannot be conformal infinities of Poincaré-Einstein metrics.
problem Existence of conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics.
method Proved the existence of infinitely many positive conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics on the ball B^8.
result Proved a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior.
New proof shows inequality without restrictions.
problem Sharp inequality relating Yamabe invariants on Poincare-Einstein manifolds.
method New proof without restrictions.
result Sharp inequality holds without restrictions.
Model for assembly map of bordism-invariant functors.
problem Understanding assembly maps of bordism-invariant functors.
method Categorical model using oplax colimits of stable, hermitian, and Poincaré categories.
result Explicit description of the kernel of the assembly map.
Study proves rigidity and gap theorems for specific metrics.
problem Existence and properties of self-dual and even Poincaré-Einstein metrics in 4D.
method Rigorous mathematical proofs, including gap theorems and rigidity results.
result Obtained new scalar conformal invariants and identified obstructions to metric existence.
We compute renormalized curvature integrals on Poincaré-Einstein manifolds.
problem Computing renormalized curvature integrals on Poincaré-Einstein manifolds.
method General procedure that connects Gauss-Bonnet-type formulas and identifies scalar conformal invariants.
result Explicit conformally invariant Gauss--Bonnet-type formulas for compact Einstein manifolds.
The delta invariant of curves on rational surfaces II: Poincaré series and topological aspectsmath.GT
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
The hyperbolic space is the only conformally compact Poincaré-Einstein manifold with a round sphere boundary.
problem Characterizing conformally compact Poincaré-Einstein manifolds.
method Optimal inequality relating relative Yamabe invariant and Yamabe invariant.
result Rigidity of the hyperbolic space as the only conformally compact Poincaré-Einstein manifold with a round sphere boundary.
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.
Poincare function counts geometric moduli, solving Arnold's conjecture.
problem Counting moduli in differential-geometric problems.
method Derives Poincare function properties and solves Arnold's conjecture.
result Derives new formulae for differential invariants and classification problems.
The paper finds manifold structures on complex spaces.
problem Constructing manifold structures on highly connected Poincaré complexes.
method Constructing examples and determining homotopy types.
result Examples of highly connected Poincaré complexes are found to be homotopy equivalent to manifolds but not smooth.
A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds. From this classification we then prove the Poincaré conjecture.
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior und…
The paper introduces Poincaré profiles for metric spaces and groups, linking them to conformal dimension.
problem Understanding the properties of metric measure spaces and groups with polynomial growth.
method Introducing and analyzing Poincaré profiles for groups and hyperbolic spaces.
result Connection between Poincaré profiles and conformal dimension, leading to non-existence of coarse embeddings.
Computes differential invariants for conformal metrics.
problem Local recognition of conformal structures.
method Computes Hilbert polynomial and Poincare function.
result Resolves the local recognition problem for conformal structures.
New polynomial distinguishes knots from the unknot.
problem No explicit knot invariant distinguishing knots from the unknot.
method Derived a Poincaré polynomial from Khovanov homology.
result First explicit knot invariant distinguishing knots from the unknot.
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.
We study a generalization of the familiar Poincaré map, first implicitely introduced by N.N. Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant tori.
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…
Expands Euler-Poincare characteristic to supergeometry.
problem No new problem introduced.
method Introduced Euler-Poincare characteristic pair in supergeometry.
result Examined transversality in -symmetric supermanifolds.
New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
New invariant connects boundary PDEs and conformal geometry.
problem Global conformal invariants of boundary PDEs.
method Variational considerations and conformal invariants.
result Compact Bach-flat manifolds with umbilic boundary admit Poincaré-Einstein metrics.
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship genera…
Nahm Poles and 0-Instantonsmath.DG
Study of 0-instantons on hyperbolic manifolds, proving invariants and energy formulas.
problem Understanding 0-instantons on hyperbolic manifolds and their properties.
method Analyzing asymptotic expansions and using Fefferman-Graham expansion for Poincaré-Einstein metrics.
result The 0-instanton obstruction tensor is a conformal invariant related to Weyl curvature, vanishing for smooth 0-instantons.
Researchers compute and describe differential invariants for self-dual conformal structures.
problem Local recognition problem for self-dual conformal structures.
method Quotient of self-duality equation, Hilbert polynomial, differential invariants, Poincaré function.
result Resolved the local recognition problem for self-dual conformal structures.
Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.
problem Computing Cheeger constants for conformally compact asymptotically constant mean curvature submanifolds.
method Analyzes conformally compact asymptotically constant mean curvature submanifolds in asymptotically hyperbolic spaces.
result Identifies conditions for Cheeger constant equality and vanishing mean curvature.
Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
problem Calculating the renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
method Decomposition of extrinsic -curvature and application to renormalized area.
result Renormalized area formula expressed as a linear combination of Euler characteristic and scalar conformal submanifold invariant.
Explains complex analytic invariants of vector fields and foliations.
problem Integrating theories of singular varieties and foliations.
method Expository discussion of invariants.
result Introduces connections between complex analytic singular varieties and foliations.
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.
Two quasi-morphisms on disk symplectomorphisms linked to Poincaré's translation number.
problem Understanding quasi-morphisms on symplectomorphism groups of the disk.
method Constructing two quasi-morphisms related to Calabi invariant and flux homomorphism.
result Relating quasi-morphisms to Poincaré's translation number.
We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie superalgebra H*(M,G) we obtain numerical smooth invariants--as opposed to homotop…
Researchers create non-normal affine structures for 3-manifolds, calculating Poincaré series.
problem Calculating Poincaré series for plumbed 3-manifolds.
method Constructing non-normal affine monoids and modules associated with negative definite plumbed 3-manifolds.
result Combinatorial formulas for Seiberg-Witten invariants and polynomial generalizations.
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.
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.
New uniform K-theory and Poincare duality established for manifolds.
problem Developing a new framework for K-theory and K-homology.
method Constructing uniform K-homology, defining external and cap products, proving homotopy invariance and Poincare duality.
result Established Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds.
We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which exp…
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.
Let be a principal G-bundle, and let be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagrangian l defined on , the bundle of connections on P.
Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.
problem Scaling symmetries in Hamiltonian systems.
method Contact reduction of symplectic Hamiltonian systems.
result Generically possible reduction to contact Hamiltonian systems, reducing inputs needed.
Homotopy invariance proven for twisted Lie algebroid cohomologies.
problem Homotopy invariance of twisted Lie algebroid cohomologies.
method Lie algebroid homotopy-invariance proof with examples.
result Comprehensive systematic way to compute twisted Lie algebroid cohomologies.
Geometric invariants of fold maps and their signatures.
problem Characterizing cobordism groups of fold maps.
method Combining immersion data and stable partial framings.
result A Poincare-Hopf type formula for the signature of manifolds.
We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We the…
Study Ricci flow on specific homogeneous spaces.
problem Behavior of Ricci flow on invariant metrics at infinity.
method Normalized Ricci flow, invariant metrics, generalized Wallach spaces, flag manifolds, Stiefel manifolds, Poincaré compactification.
result Characterized behavior of Ricci flow for specified homogeneous spaces.
Study positive scalar curvature on manifolds with group actions, proving rigidity and duality results.
problem Obstructions and existence of positive scalar curvature metrics on manifolds with group actions.
method G-equivariant index theory for proper co-compact actions, rigidity results for almost-complex manifolds, Poincaré duality.
result Established Poincaré duality between G-equivariant K-homology and K-theory for almost-connected Lie groups or discrete groups.
This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficien…
The paper re-evaluates eigenvalue estimates and rigidity of Poincare-Einstein metrics.
problem Eigenvalue estimates and rigidity of Poincare-Einstein metrics on spin manifolds.
method Revisits eigenvalue estimates of the Dirac operator and proves rigidity results under weaker conditions.
result Poincaré-Einstein metrics are rigid under specific eigenvalue conditions.