3D foliation study finds Poisson structure with specific singularities.
problem Characterizing Poisson structures on Bott-Morse foliations in 3D.
method Analyzes Bott-Morse foliations, computes Poisson bivectors and symplectic forms.
result Linear, singular Poisson structure of rank 2 with Bott-Morse singularities found.
We study codimension one (transversally oriented) foliations $\fa$ on oriented closed manifolds M having non-empty compact singular set $\sing(\fa)$ which is locally defined by Bott-Morse functions. We prove that if the transverse type of $\fa$ at each singular point is a center and $\fa$ has a compact leaf with fini…
We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among…
Research resolves sign conventions in Floer theory for Morse-Bott case.
problem Sign conventions in filtered A∞-operations for Lagrangian Floer theory. method Defined filtered A∞-operations and verified formulae using de Rham model. result Resolved sign issues in Bott-Morse setting.
We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are co…
Researchers extend Higgs bundle theory to parabolic bundles, counting components.
problem Counting components in moduli spaces of Higgs bundles with parabolic structures.
method Generalized Beauville-Narasimhan-Ramanan correspondence, Bott-Morse theory.
result Exact component count for maximal parabolic Sp(2n,ℝ)-Higgs bundles.
We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott--Morse theory on the corresponding moduli spaces.
We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the spl…
In this paper, we consider generalized moment maps for Hamiltonian actions on H-twisted generalized complex manifolds introduced by Lin and Tolman \cite{Lin}. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact H…
Study on minimal foliations in 3D manifolds with specific conditions.
problem Characterizing minimal foliations in 3D manifolds.
method Analyzing Anosov foliations and their intersections.
result Necessary and sufficient conditions for orbit foliation of Anosov flows.
We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…
Homogeneous three-spheres have only homogenous foliations.
problem Characterize foliations of homogeneous three-spheres.
method Prove that a three-sphere's metric foliations are homogenous if and only if it is naturally reductive.
result Homogeneous three-spheres have only homogenous foliations.
Integral volume vanishes for manifolds with circle foliations.
problem Integral foliated simplicial volume calculation.
method Regular foliation by circles analysis.
result Integral foliated simplicial volume vanishes.
Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.
problem Investigate foliations transverse to closed conformal vector fields on Riemannian manifolds.
method Analyze conditions for totally geodesic leaves and geometric constraints on foliations.
result Characterize totally geodesic foliations and classify minimal and constant mean curvature foliations.
We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz coh…
The study limits the number of specific foliations with bounded geometry.
problem Bounding the number of isoparametric foliations with bounded geometry.
method Proving finitely many foliations with specific properties and constructing infinite families of non-diffeomorphic foliations.
result There are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, up to foliated diffeomorphism.
Develops deformation theory for symplectic foliations using L∞-algebras.
problem Deformation of symplectic foliations.
method Uses L∞-algebras to control deformation problems. result Establishes a correspondence between small deformations and Maurer-Cartan elements of L∞-algebra. The paper examines a modified Godbillon-Vey class for Reeb foliations and finds it non-trivial for some foliations.
problem Characterizing foliations using the modified Godbillon-Vey class.
method Defined and analyzed the modified Godbillon-Vey class for Reeb foliations.
result The modified Godbillon-Vey class can distinguish non-diffeomorphic foliations and is non-trivial for some foliations.
Simple flows on manifold foliations.
problem Transversely oriented foliations on closed manifolds.
method Simple foliated flows on codimension one.
result Existence of simple foliated flows on manifolds.
Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and (F,F′)f-harmonic maps. result Equivalence of transversally f-harmonic and (F,F′)f-harmonic maps in minimal foliations. Survey on Killing foliations with technical advantages.
problem Understanding closures of Riemannian foliations.
method Review of Molino's structural theory and transverse isometry theory.
result Closures of Killing foliations described by transverse Killing vector fields.
Proves conjecture about foliations on curved spaces.
problem Completeness of dual foliations on curved spaces.
method Analyzes Riemannian foliations on nonnegatively curved symmetric spaces.
result Foliations split into trivial and single dual leaf foliations.
Simplified proof of foliation closure theorem for linear foliations.
problem Proving the closure of linear foliations on Riemannian manifolds.
method Direct geometric approach, focusing on projectable foliations and compatible connections.
result Smoothness of the closure of linear foliations directly proven.
Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
problem Characterizing Lie foliations with symmetric leaves.
method Analyzing the rigidity of Lie foliations with locally symmetric leaves.
result Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
New foliations constructed from contact pairs, revealing flexible taut foliations.
problem Characterizing taut foliations in 3D.
method Construction of codimension-one foliations from pairs of contact structures in 3D.
result Foliations constructed are taut, providing new insights into the L-space conjecture.
The paper resolves singular foliations through a series of blowups.
problem Singular foliations that cannot be resolved directly.
method Applying Nash modifications to the universal Lie ∞-algebroid of a singular foliation.
result Any singular foliation becomes a Debord foliation after one blowup.
Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
This paper compares two invariants of foliated manifolds which seem to measure the non-Hausdorffness of the leaf space: the transversal length on the fundamental group and the foliated Gromov norm on the homology. We consider foliations with the property that the set of singular simplices transverse to the foliation sa…
Classifies neighborhoods around specific leaf structures.
problem Classifying singular foliations with given leaf and transverse singular foliation.
method Analyzes the structure of singular foliations and their leaves.
result Developed a method to classify neighborhoods around specific leaf structures.
Uniform foliations with Reeb components on 3-manifolds.
problem Characterizing uniform foliations with Reeb components.
method Study of foliations on 3-manifolds with infinite fundamental group.
result Examples and results on behavior of foliations.
Survey and extend work on singular foliations in diffeology.
problem Understanding singular foliations and their properties in diffeological settings.
method Survey Stefan and Sussmann's work, introduce transverse equivalence, and define basic cohomology.
result Transverse equivalence of singular foliations preserves leaf spaces diffeologically but not conversely.
The paper proves a theorem about completely solvable Lie foliations on compact manifolds.
problem Generalizing Haefliger's theorem to completely solvable Lie foliations.
method Proves that every completely solvable Lie foliation on a compact manifold is the inverse image of a homogeneous foliation.
result Every completely solvable Lie foliation on a compact manifold is the inverse image of a homogeneous foliation.
We extend the Eliashberg-Thurston theorem on approximations of taut oriented C2-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented C1,0-foliations, where by C1,0 foliation, we mean a foliation with continuous tangent plane field. These C1,0-fol…
Tight contact foliations can be made less restrictive.
problem Homotopy invariance of tightness in contact foliations.
method Turbulisation procedure to make tight foliations less restrictive.
result Tightness is not a homotopy invariant property.
Classifies foliations on CROSSes.
problem Classifying Riemannian foliations on CROSSes.
method Classification based on manifold homeomorphism.
result Classification of Riemannian foliations on CROSSes.
Study on foliation automorphisms, finding non-Lie groups and ILH Lie groups.
problem Understanding the structure of diffeomorphism groups of foliations.
method Investigation of diffeomorphism groups of foliations, proving properties of automorphism groups.
result Found examples of foliations with non-Lie automorphism groups and proved properties of ILH Lie groups for certain foliations.
The paper introduces foliated open books for contact 3-manifolds with boundary foliations.
problem Studying contact manifolds with convex boundary using finer tools.
method Developed a new type of open book decomposition with a specified characteristic foliation on the boundary.
result Established the uniqueness and existence of foliated open books and their equivalence to other models.
Study on Morse foliations on S^3, proving instability.
problem Existence and instability of Morse foliations on S^3.
method Analysis of Morse foliations of codimension one on S^3.
result Proven instability of Morse foliations in almost all cases.
The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an A∞ de Rham theorem and constructing an integration functor. result An equivalence between ∞-representations of L∞-algebroids and ∞-representations of Lie ∞-groupoids for foliations. Develops twistor theory for foliated manifolds, proving orbifold results.
problem Classical twistor theory applied to foliated manifolds.
method Constructs twistor space of normal bundle, proves foliated versions of results.
result Obtains orbifold versions of classical results.
No projective structure found on foliations of elliptic curves.
problem Existence of transversely projective structures on foliations.
method Analyzing foliations on the product of two elliptic curves.
result Generic nonsingular turbulent foliations do not have transversely projective structures.
Classifies polar foliations on symmetric spaces.
problem Classifying polar foliations on symmetric spaces.
method Orbit equivalence and classification up to codimension two.
result Foliations are either hyperpolar or extensions of rank one foliations.
Extends symplectic flow results to foliations.
problem Applying hard Lefschetz theorem to symplectic foliations.
method Generalizes results from symplectic flows to foliations.
result Extends transversal hard Lefschetz theorem to transversely symplectic foliations.
Positive curvature forces foliation leaf spaces to have boundaries.
problem Understanding boundaries in foliated leaf spaces with positive curvature.
method Analyzing singular Riemannian foliations with positive sectional curvature.
result Polar foliations of positively curved manifolds have leaf spaces with nonempty boundaries.
Introduces singular Finsler foliations and their properties.
problem Generalizing Finsler actions, submersions, and foliations.
method Definition and analysis of singular Finsler foliations on Randers manifolds.
result Singular Finsler foliations on Randers manifolds are singular Riemannian foliations on the Riemannian manifold induced by the metric.
Proves a Thom isomorphism for foliated differential forms.
problem Transverse Lie algebra actions on foliated manifolds and vector bundles.
method Thom isomorphism theorem for differential forms in foliated settings.
result Established a new Thom isomorphism theorem.
A foliation of a manifold M is called R-covered if its lift to the universal cover of M has space of leaves R. We show that there are many graph manifolds which admit taut foliations, but which do not admit any R-covered foliations. On the other hand, we show that these manifolds all have finite covers admitting R-cove…
Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger coh…