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 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. 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.
The paper establishes a correspondence between normal distributions and neat foliations on manifolds with boundary.
problem Understanding normal distributions on manifolds with boundary.
method Develops a theory analogous to Stefan and Sussmann's for integrable distributions, focusing on neat foliations.
result A one-to-one correspondence between neatly integrable normal distributions and neat foliations by manifolds with boundary.
Research shows conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.
problem Conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.
method Analyzes the geometry of the domain's boundary to determine foliation conditions.
result Conditional foliation is possible but not guaranteed, depending on the domain's geometry.
Let M be a fibered 3-manifold with multiple boundary components. We show that the fiber structure of M transforms to closely related transversely oriented taut foliations realizing all rational multislopes in some open neighborhood of the multislope of the fiber. Each such foliation extends to a taut foliation in t…
The paper proves a foliation of a manifold with specific properties.
problem Proving a foliation with constant mean curvature and perpendicular boundary condition.
method Analyzing a Riemannian manifold with smooth boundary and proving the existence of a foliation.
result Existence of a smooth foliation around a nondegenerate critical point of the mean curvature function of the boundary.
We define the Analytical signature, the Hodge signature and the de Rham signature for a foliated manifold with boundary with foliation transverse to the boundary. We show that all these signatures coincide and a Hirzebruch formula is valid.
The paper proves properties of boundaries of Riemannian foliations.
problem Proving properties of boundaries of Riemannian foliations.
method Proved boundaries of orbit spaces and leaf spaces are Alexandrov spaces with lower curvature bounds.
result Rigidity theorem for positively curved leaf spaces with maximal boundary volume.
We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative inde…
The paper studies diffeomorphisms of a specific foliation on a Klein bottle.
problem Computing homotopy types of diffeomorphism groups for a specific foliation.
method Analyzes a Morse-Bott foliation on a solid Klein bottle and its twisted bundle.
result Computes homotopy types of foliated and leaf-preserving diffeomorphism groups.
Characterizes surfaces made from strips with boundary leaves.
problem Classifying foliated surfaces formed from strips.
method Analyzes surfaces (Z,Δ) glued from open strips with boundary leaves. result Characterizes a subclass of foliated surfaces.
Persistently foliar knots have a unique foliation property under certain surgeries.
problem Characterizing knots with a specific foliation property under surgeries.
method Analyzing foliations in knot complements and using Dehn surgery.
result Composite knots and nontrivial connected sums of fibered knots are persistently foliar.
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.
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.
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.
We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the cl…
Study shows rigidity for spin^c manifolds with foliated boundaries.
problem Rigidity of Riemannian spin^c manifolds with foliated boundaries.
method Integral inequality based on geometric quantities.
result Characterization of equality cases for specific manifolds.
The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…
Integral foliated simplicial volume vanishes for manifolds with S1-action.
problem Integral foliated simplicial volume of manifolds with S1-action. method Geometric construction of Yano's proof for ordinary simplicial volume combined with parametrised uniform boundary condition for S1. result Integral foliated simplicial volume of aspherical manifolds with S1-action vanishes. Let X be a connected non-compact 2-dimensional manifold possibly with boundary and Δ be a foliation on X such that each leaf ω∈Δ is homeomorphic to R and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. H…
We study the behavior of the Yang-Mills flow for unitary connections on compact and non-compact oriented surfaces with varying metrics. The flow can be used to define a one dimensional foliation on the space of SU(2) representations of a once punctured surface. This foliation universalizes over Teichmüller space and is…
For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of L2 harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the pr…
The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…
The paper explores universal circles for Anosov foliations and their uniqueness.
problem Exploring the uniqueness of universal circles for Anosov foliations.
method Using the flow space of an Anosov flow to parameterize the circle bundle at infinity of the foliations.
result Several constructions of a universal circle are typically distinct and not conjugate.
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.
An alternate proof shows how foliation extensions work in 3D spaces.
problem Continuous extension of foliations in 3D spaces.
method Uses the universal circle and properties of pseudo-Anosov flows.
result Shows how all continuous extensions organize in the boundary.
We study non-compact surfaces obtained by gluing strips R×(−1,1) with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the gro…
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple descripti…
The paper studies quasifuchsian manifolds and their boundary foliations, providing formulas and extensions.
problem Understanding the boundary behavior of quasifuchsian manifolds and their foliations.
method Variation formula for renormalized volume, upper bound on extremal length, extensions of quadratic differential.
result Upper bound on extremal length of horizontal measured foliation and extensions of quadratic differential.
Let X0 be a compact Riemannian manifold with boundary endowed with a oriented, measured even dimensional foliation with purely transverse boundary. Let X be the manifold with cylinder attached and extended foliation. We prove that the L2--measured index of a Dirac type operator is well defined and the following…
We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique cons…
Study shows universal circle isomorphic to flow space ideal boundary.
problem Understanding foliations and pseudo-Anosov flows in 3-manifolds.
method Analyzing depth-one foliations and pseudo-Anosov flows in atoroidal 3-manifolds.
result Universal circle is isomorphic to flow space ideal boundary.
3-manifolds foliar if surfaces minimize Thurston norm, leading to taut foliations.
problem Understanding foliar 3-manifolds via minimizing surfaces.
method Defining double-diamond taut sutured manifolds and analyzing their implications.
result Many slopes are strongly realized in foliations of 3-manifolds, including knot complements.
Study of diffeomorphisms groups on lens spaces with Morse-Bott foliations.
problem Computing homotopy types of diffeomorphism groups of specific foliations.
method Analysis of leaf-preserving and foliated diffeomorphisms on lens spaces.
result Inclusion of leaf-preserving groups into foliated groups is a homotopy equivalence.
Study foliations of hyperbolic 3-manifolds with constant Gaussian curvature.
problem Understanding foliations of hyperbolic 3-manifolds with constant Gaussian curvature.
method Analyzing Thurston and Schwarzian parametrizations, proving Kleinian reciprocity, and describing foliations as Hamiltonian vector fields.
result Generalization of McMullen's Kleinian reciprocity theorem and description of constant curvature foliations.
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.
Paper shows ergodicity and irreducibility of mapping class group boundary representation.
problem Ergodicity and irreducibility of mapping class group boundary representation.
method Statistical hyperbolicity and classical result of Masur generalization.
result Boundary representation of mapping class group is ergodic and irreducible.
Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to …
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
Contractible diffeomorphism groups on lens spaces derived from Morse-Bott foliations.
problem Understanding the structure of diffeomorphism groups on lens spaces.
method Analyzing Morse-Bott foliations and their diffeomorphisms.
result Contractible diffeomorphism groups on lens spaces.
We develop a method for preserving pseudoholomorphic curves in contact 3-manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3-manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1-parameter fam…
Innovative contact invariant derived from Heegaard Floer homology.
problem Contact structures in three-manifolds with boundary.
method Developed a contact invariant using bordered sutured Heegaard Floer homology.
result Identified well-defined classes in bordered sutured modules.
The authors introduce Morse foliated open books for studying contact manifolds.
problem Studying contact manifolds with boundary.
method Introducing Morse foliated open books and extending the right-veering concept.
result Right-veering plays a similar role in detecting overtwistedness in foliated open books.
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.
problem Determining foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus.
method Inspired by Bonahon's method, uses measured bending laminations on the boundary of convex cores.
result Measured foliations at infinity of quasi-Fuchsian manifolds can be uniquely realized for small t. A taut foliation of a hyperbolic 3-manifold has the continuous extension property for leaves in almost every direction; that is, for each leaf of the universal cover of the foliation and almost every geodesic ray in the leaf, the limit of the ray in the universal cover of the 3-manifold is a well-defined point in the i…
In this paper, we define lower dimensional volumes associated to sub-Dirac operators for foliations. In some cases, we compute these lower dimensional volumes. We also prove the Kastler-Kalau-Walze type theorems for foliations with or without boundary. As a corollary, we give an explanation of the gravitational action …