The study characterizes pseudo-Riemannian foliations and their graphs.
problem Characterizing pseudo-Riemannian foliations and their graphs.
method Analyzing geodesics and metrics on foliation graphs.
result A unique pseudo-Riemannian metric exists on foliation graphs.
Graph manifolds without smooth foliations are not L-spaces.
problem Characterizing graph manifolds that are not L-spaces.
method Extending gluing theorem to graph manifold rational homology solid tori and applying to foliations.
result Graph manifolds with left-orderable fundamental groups are not L-spaces.
Graph manifolds admit taut foliations unless they are L-spaces.
problem Characterizing graph manifolds with taut foliations.
method Using properties of L-spaces and left-orderability.
result Graph manifolds are L-spaces if and only if their fundamental group is not left-orderable.
Quadratic differentials on punctured surfaces link foliations and metric graphs.
problem Understanding the structure of quadratic differentials on punctured surfaces.
method Introducing asymptotic directions and analyzing foliations and metric graphs.
result A unique meromorphic quadratic differential can be constructed for any prescribed horizontal foliation.
We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-…
The paper solves a curvature equation for a specific spacetime foliation.
problem Characterizing foliations in a Generalized Robertson-Walker spacetime.
method Analyzing a mean curvature equation on spacelike graphs.
result All entire solutions of the curvature equation are found.
The topology of the Hausdorff leaf spaces (HLS) for a codim-1 foliation is the main topic of this paper. At the beginning, the connection between the Hausdorff leaf space and a warped foliations is examined. Next, the author describes the HLS for all basic constructions of foliations such as transversal and tangential …
We present an equivalent criterion for the global existence of Euler's multiplier for an integrable one-form taking into account the corresponding codim-1-foliation. In particular, the impact of inseparable leaves is considered. Here, we suppose that the foliation can be reduced to a graph; we also discuss obstructions…
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…
Study classifies singular foliations on complex plane, finding finite moduli space.
problem Classifying singular foliations on complex plane.
method Fixed topological invariants and computed moduli space using cohomology of group-graph.
result Proved moduli space has finite dimension under generic conditions.
A formula calculates the Euler class of foliations using dual graphs.
problem Calculating the Euler class of foliations using cooriented branched surfaces.
method Using dual graphs of cooriented branched surfaces to define a simplicial 1-cycle representing the Poincaré dual of the Euler class.
result The formula generalizes previous results and classifies realizable homology classes.
The study explores translating solitons in semi-Riemannian foliations and provides criteria for their existence.
problem Understanding and characterizing translating solitons in semi-Riemannian foliations.
method Recalling and extending the concept of translating solitons, analyzing submersions, and employing criteria for lifting or projecting solitons.
result Explicit criteria for constructing translating solitons in foliated semi-Riemannian manifolds.
We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of C2-functions to leaves of transversally oriented codimension one C2-foliations of Riemannian manifolds. That extends partially Salavessa's work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-O…
Study the Gromov boundary of fine curve graph for surface homeomorphisms.
problem Understanding the boundary of fine curve graph for surface homeomorphisms.
method Examined the Gromov boundary and local topology near specific foliations and laminations.
result Found elements with positive stable commutator length and proved a Tits alternative.
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect fa…
The paper explores holonomy, zeta functions, and cohomology in foliated manifolds with stratified boundaries.
problem Understanding symmetries and cohomology in foliated manifolds with stratified boundaries.
method Developed a novel formalism for the Gamma-set and defined an Ihara zeta function to encode symmetries. Investigated the relationship between holonomy and zeta functions, and analyzed how the twist map impacts cohomology.
result Conjectured a duality between holonomy fixed points and the poles of the Ihara zeta function, extending to twisted cohomology classes.
Graph manifold L-space intervals computed and applied to cables.
problem Computing L-space intervals for graph manifolds and their cables.
method Graph manifold analog of Jankins-Neumann classification, Floer simple manifolds generalization.
result Finite recursive formula for L-space intervals of graph manifolds and cable knots.
Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces CP^n are classified, except if n=15 and q=1. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations…
The paper studies the homeotopy groups of foliations on surfaces.
problem Understanding the homeotopy groups of foliations on surfaces.
method Analyzing the quotient of homeomorphisms of a foliation group by its identity component.
result Identifies the quotient group with automorphisms of a graph encoding foliation combinatorics.
Unique foliation of AdS3 domains by constant mean curvature surfaces.
problem Foliation of domains of dependence in AdS3. method Proving existence and uniqueness of constant mean curvature foliation.
result Existence and uniqueness of foliation by constant mean curvature surfaces.
We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexis…
The paper studies homeotopy groups of leaf spaces for specific foliations.
problem Identifying homeotopy groups of leaf spaces for non-compact surfaces with non-compact leaves.
method Identifying homeotopy groups with automorphisms of graphs and showing induced homomorphisms.
result The induced homomorphism between homeotopy groups is either injective or has a kernel of Z_2.
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 shows saddle connection graph's geometry and quasi-isometry properties.
problem Characterize the geometry and quasi-isometry of saddle connection graphs.
method Proved 4-hyperbolicity and uniform quasi-isometry to a tree, used generalised unicorn paths.
result Saddle connection graph is not quasi-isometrically rigid and its boundary is straight foliations.
Critical graphs of quadratic differentials equidistribute in moduli space.
problem Distribution of critical graphs in moduli space.
method Study of Jenkins-Strebel differentials and their critical graphs.
result Critical graphs equidistribute to the Kontsevich measure.
By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to…
Toroidal 3-manifolds have special group structures that can be shown through specific covers.
problem Characterizing the fundamental groups of toroidal 3-manifolds.
method Proving toroidal 3-manifolds have circularly-orderable fundamental groups by showing they admit finite cyclic covers with left-orderable fundamental groups.
result Toroidal 3-manifolds have circularly-orderable fundamental groups, which can be shown through specific covers.
New method shows pseudo-Anosov flows on graph manifolds can be simplified.
problem Understanding pseudo-Anosov flows on graph manifolds.
method Constructing a partial Birkhoff section with genus one components that misses finitely many closed orbits.
result Every pseudo-Anosov flow on a graph manifold is almost equivalent to a totally periodic flow or a suspension Anosov flow.
We show how to construct, for each r≥3, an ageometric, fully irreducible φ∈Out(Fr) whose ideal Whitehead graph is the complete graph on 2r−1 vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal …
Survey uses Milnor fibrations to classify first integrals of differential systems.
problem Classifying first integrals of differential systems using geometric-topological methods.
method Utilizing Milnor fibrations and connections with harmonic morphisms to provide topological and geometric descriptions.
result Geometric-topological classifications of first integrals for both isolated and non-isolated singularities.
We give a new proof of the existence (\cite{HM}, \cite{Ren}) of a Jenkins-Strebel differential Φ on a Riemann surface $\SR$ with prescribed heights of cylinders by considering the harmonic map from $\SR$ to the leaf space of the vertical foliation of Φ, thought of as a Riemannian graph. The novelty of the argument …
Geometric non-commutative geometry proves non-existence of certain metrics.
problem Proving non-existence of metrics of positive scalar curvature on foliations.
method Detailed review and extension of existing results, including new obstructions.
result Extension of non-existence results to non-compact manifolds of bounded geometry.
This is a survey concerning the relationship between Lie Groupoids (and their morphisms) and singular foliations in the sense of Sussmann-Stefan (considered from a purely geometrical point of view). We focus on the interaction between the algebraic and differentiable structures underlying Lie groupoids, and between gro…
Researchers calculate the volume of Seifert representations for graph manifolds and their covers.
problem Computing the volume of Seifert representations for graph manifolds and their finite covers.
method Established an effective formula for computing the volume of Seifert representations of graph manifolds and obtained restrictions analogous to the Milnor–Wood inequality.
result The Seifert volume of any graph manifold is a rational multiple of π², and the supremum ratio of the Seifert volume over the covering degree can be positive or infinite.
The paper proves foliations of solutions to the minimal surface equation in exterior domains.
problem Existence and properties of foliations by solutions to the exterior Dirichlet problem for minimal surfaces.
method Analyzes a 1-parameter family of solutions to the minimal surface equation in exterior domains with specific boundary conditions.
result Foliation of the open subset in R^(n+1) by graphs of solutions, with bounds and asymptotic behavior.
We consider spacelike graphs Γf of simple products (M×N,g×−h) where (M,g) and (N,h) are Riemannian manifolds and f:M→N is a smooth map. Under the condition of the Cheeger constant of M to be zero and some condition on the second fundamental form at infinity, we conclude that if $Γ_f \subset…
We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler cha…
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.
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solution…
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.
Proves geometric invariance of signature and cohomology for Riemannian foliations.
problem Defining and proving invariance of geometric invariants for Riemannian foliations.
method Analyzes basic signature and Lichnerowicz cohomology under homotopy equivalence.
result Foliated homotopy invariance of basic signature and cohomology.
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.
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.
Complete classification of foliations on spheres from Clifford systems.
problem Classifying foliations on spheres from Clifford systems.
method Classification of homogeneous singular Riemannian foliations of spheres.
result Classification completed for foliations initiated by the second author.
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.