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.
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.
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.
Classifies foliations on CROSSes.
problem Classifying Riemannian foliations on CROSSes.
method Classification based on manifold homeomorphism.
result Classification of Riemannian foliations on CROSSes.
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…
Constructs foliations from manifold group left orders.
problem Creating foliations from group structures.
method Inputting a left order of a manifold's fundamental group produces a foliation.
result Establishes connections to 3D problems.
Study on transverse Ricci solitons on compact foliated manifolds.
problem Characterizing transverse Ricci solitons on compact foliated manifolds.
method Investigation of self-similar solutions of the transverse Ricci flow, analysis of taut Riemannian foliations.
result Established relations between taut Riemannian foliations and transverse Ricci solitons, found examples of transverse Ricci solitons.
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.
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…
The study explores conformal symplectic foliations on closed manifolds, proving their existence in dimensions 5 and above.
problem Existence of conformal symplectic foliations on closed manifolds.
method Analysis of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5.
result Construction of conformal symplectic foliations on closed, simply-connected, almost contact manifolds in dimension 5.
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.
The paper studies affine manifolds with linear foliations and their topological properties.
problem Characterizing the topology of affine manifolds with linear foliations.
method Analyzing the structure of affine manifolds and their foliations, proving topological properties.
result Compact affine manifolds with linear foliations are homeomorphic to the torus under certain conditions.
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.
Solves generalized Kazdan-Warner equations on foliated manifolds.
problem Existence and uniqueness of solutions to generalized Kazdan-Warner equations on foliated manifolds.
method Extends theorem to compact foliated manifolds, provides examples of PDEs.
result Solves the transverse Hitchin equation and its generalizations.
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.
We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we c…
The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.
problem Solving foliation singularities on Sasakian 5-manifolds.
method Applying the Sasaki-Ricci flow to resolve cyclic quotient foliation singularities.
result Proves a Sasaki analogue of the analytic minimal model program.
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.
The article derives integral formulas for foliated sub-Riemannian manifolds.
problem Integrating geometric concepts in Riemannian manifolds with foliations.
method Deriving integral formulas involving shape operators and curvature tensor.
result Generalizes results for foliated Riemannian manifolds and includes arbitrary functions.
The paper develops L2 theory for foliations on manifolds with boundary.
problem Analyzing the cohomology of foliated manifolds with boundary.
method Develops L2 theory, establishes decomposition and vanishing theorems, and proves duality and extension theorems. result Establishes Dolbeault decomposition of basic forms and proves global regularity for ∂ˉB-equations. 3-manifolds with positive scalar curvature have controlled foliations.
problem Understanding foliations in 3-manifolds with positive scalar curvature.
method Showed a singular foliation by surfaces with controlled area and diameter.
result 3-manifolds with positive scalar curvature admit controlled foliations.
Survey on holomorphic structures on complex manifolds.
problem Holomorphic foliations with transverse holomorphic Cartan geometries.
method Analyzes G-structures and Cartan geometries on compact complex manifolds.
result Foliated holomorphic Cartan geometries on compact complex manifolds.
Study cohomology of quaternionic foliations and orbifolds.
problem Understanding cohomology of quaternionic foliations and orbifolds.
method Definition and proof of foliated versions of classical results for quaternionic Kähler manifolds.
result Formulation and proof of foliated versions of classical results for quaternionic Kähler manifolds.
Study shows how to deform foliated manifolds with flat leaves, leading to geometric characterization.
problem Deforming foliated manifolds with flat leaves while maintaining curvature bounds.
method Collapsing a manifold with a closed flat regular Riemannian foliation, keeping curvature uniformly bounded.
result For compact, simply connected manifolds, foliations are given by torus actions.
Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…
The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.
problem Existence of monotone CMC foliations for quasi-Fuchsian manifolds.
method Analyzes quasi-Fuchsian manifolds near the Fuchsian locus and proves the existence of a unique monotone CMC foliation.
result Proves the existence of a unique monotone CMC foliation for quasi-Fuchsian manifolds in a small neighborhood of the Fuchsian locus.
Left orderability proven for certain 3-manifolds with specific foliations.
problem Proving left orderability for specific 3-manifolds with taut foliations.
method Analyzing the fundamental group of a 3-manifold with a co-orientable taut foliation.
result The fundamental group of the 3-manifold is left orderable.
A K(pi,1)-foliation is one for which the universal covers of all leaves are contractible (thus all leaves are K(pi,1)'s for some pi). In the first part of the paper we show that the tangential Lusternik--Schnirelmann category cat F of a K(pi,1)-foliation F on a manifold M is bounded from below by t-codim F for any t wi…
We give a complete classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient manifold. The results are analogues of Haefliger's classification of foliations on open manifold.
Introduces positivity for (1,1) classes in foliated manifolds.
problem Understanding positivity in foliated manifolds.
method Introduces positivity for a real basic (1,1) class in basic Bott-Chern cohomology group and studies its relationship with negativity of transverse holomorphic sectional curvature. result Establishes the relationship between positivity and negativity in foliated manifolds.
We investigate contrasting behaviours emerging when studying foliations on non-metrisable manifolds. It is shown that Kneser's pathology of a manifold foliated by a single leaf cannot occur with foliations of dimension-one. On the other hand, there are open surfaces admitting no foliations. This is derived from a quali…
3-manifolds with Heegaard 2 admit taut foliations if their fundamental group is left-orderable.
problem Characterizing 3-manifolds with taut foliations.
method Proving existence of taut foliations based on left-orderability of the fundamental group.
result 3-manifolds with Heegaard genus 2 and left-orderable fundamental group admit co-orientable taut foliations.
We study transversely Lorentzian foliations on the closed 3-manifolds. We classify them under a completeness hypothesis and we deduce the dual classification of codimension 1 geodesically complete timelike totally geodesic foliations. Besides we provide an example of a Lorentzian foliation on a compact 3-manifold which…
In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterizat…
Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.
problem Understanding foliations at the boundary of quasi-Fuchsian manifolds.
method Proving filling pairs and showing realisation of measured foliations.
result Proved that measured foliations at infinity of quasi-Fuchsian manifolds are filling when close to being Fuchsian.
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
New examples of non-homeomorphic foliation leaves found.
problem Finding non-homeomorphic foliation leaves in manifolds.
method Examples of 5-manifolds and foliations of 6-manifolds.
result Examples of non-homeomorphic foliation leaves in C1 and C∞ foliations. The study examines spacelike foliations on Lorentz manifolds under specific conditions.
problem Investigating geometric properties of spacelike foliations on Lorentz manifolds.
method Analyzing conditions for stability, total geodesy, and total umbilicity of foliation leaves.
result Conditions for the stability, total geodesy, and total umbilicity of spacelike foliation leaves are established.
Study on tautness tensor for Riemannian foliations.
problem Understanding tautness properties of Riemannian foliations.
method Investigating a symmetric 2-tensor related to mean curvature.
result Prove a tautness condition for compact manifolds.
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.
Torus decomposition shows foliation detected slopes for glued knot manifolds.
problem Detecting foliation detected slopes in glued knot manifolds.
method Torus decomposition and foliation analysis.
result Gluing knot manifolds identifies rational boundary slopes.
Defines basic Albanese maps for foliated Riemannian manifolds.
problem No specific problem stated; focuses on new concept definition.
method Introduces basic Albanese map using basic 1-forms.
result Relates basic Albanese map to classical Albanese map.
We produce examples of taut foliations of hyperbolic 3-manifolds which are R-covered but not uniform --- ie the leaf space of the universal cover is R, but pairs of leaves are not contained in bounded neighborhoods of each other. This answers in the negative a conjecture of Thurston `Three-manifolds, foliations and cir…
The paper studies a new structure on contact manifolds and its foliation properties.
problem Investigating a new structure on contact manifolds.
method Examining weak nearly Sasakian structures and their foliations.
result The weak nearly Sasakian structure foliates into two types of totally geodesic foliations.
Survey on collapsing manifolds using group actions and foliations.
problem Collapsing manifolds with controlled curvature.
method Using group actions and singular Riemannian foliations.
result Recent extensions to singular Riemannian foliations.
Study constructs non-funnel foliations in 3D manifolds.
problem Does the funnel property follow from leafwise quasigeodesic foliations?
method Constructs 1D foliations within 2D subfoliations in 3-manifolds.
result Not all quasigeodesics share a common ideal point in most leaves.
Compact foliations preserve entropy if leaves are strictly convex projective.
problem Entropy rigidity for foliations by strictly convex projective manifolds.
method Analysis of foliated volume entropies and homeomorphisms.
result Equality in foliated volume entropies implies homothetic leaves.
We define Seiberg-Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K-contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and …