Smooth algebra analysis for one-dimensional singular foliations.
problem Analyzing smooth algebras of one-dimensional singular foliations.
method Analyzing natural ideals and using Dixmier-Malliavin theorem.
result Smooth algebras of one-dimensional singular foliations are pairwise nonisomorphic.
The paper explores global index formulas for one-dimensional holomorphic foliations.
problem Global index formulas for one-dimensional holomorphic foliations.
method Microlocal point of view and short proofs for existing index formulas.
result Generalizations of existing index formulas.
The paper proves residue formulas for logarithmic foliations on non-compact manifolds.
problem Analyzing logarithmic foliations on non-compact complex manifolds.
method Proves Baum-Bott type formula for residue.
result Provides a Poincaré-Hopf type theorem and optimal description for foliations.
Study examines conditions for quotient maps of foliated manifolds to be locally trivial.
problem Conditions for quotient maps of foliated manifolds to be locally trivial.
method Analyzes necessary and sufficient conditions for a quotient map to be a locally trivial fibration.
result Necessary and sufficient conditions for the map to be a locally trivial fibration are presented.
Defines and calculates foliation homology from flows.
problem Homology of foliations defined by flows.
method Definition and calculation of foliation homology.
result Homology naturally associated with Seifert fibration.
We extend the unpublished work of M. Handel and R. Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel-Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic lamaniations, show the geodesic laminations satisfy the axioms, and p…
We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of foliated manifold with leafwise Randers structure. In this case the estimate…
Uniform bounds on center leaves volume for codimension one center foliations.
problem Bounding the volume of center leaves in codimension one center foliations.
method Analyzing dynamically coherent partially hyperbolic diffeomorphisms with one-dimensional unstable bundle.
result Volume of center leaves is uniformly bounded.
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.
Anosov flow found in specific partially hyperbolic systems.
problem Characterizing partially hyperbolic diffeomorphisms with center foliation.
method Analyzing transitive dynamically coherent systems with one-dimensional center foliation.
result Discretized Anosov flow found in systems satisfying f(W)=W for center leaves. A flow of metrics, gt, on a manifold is a solution of a differential equation $\dt g = S(g)$, where a geometric functional S(g) is a symmetric (0,2)-tensor usually related to some kind of curvature. The mixed sectional curvature of a foliated manifold regulates the deviation of leaves along the leaf geodesics. W…
Defines Milnor number for foliations and shows its topological invariance.
problem Defining and proving invariance of Milnor number for non-isolated singularities of holomorphic foliations.
method Defining Milnor number as intersection number of sections; proving invariance via C1 topological equivalences. result Milnor number is invariant under C1 topological equivalences. Enhances foliation results for non-C2 codimension one foliations.
problem Classical foliation results for non-C2 foliations. method Flow box decompositions and enhancements of classical results.
result Extensions of foliation results to non-C2 foliations. Formula for foliations' singularities in complex projective spaces.
problem Counting singularities of foliations on complex projective spaces.
method Global residue formula for logarithmic indices of foliations with isolated singularities.
result Formula for the number of singularities in the complement of the invariant divisor on complex projective spaces.
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…
Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center…
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.
New findings on leafwise quasi-geodesic foliations in 3-manifolds.
problem Understanding leafwise quasi-geodesic foliations in 3-manifolds.
method Analyzing intersections of transverse foliations in 3-manifolds with Gromov hyperbolic leaves.
result Hausdorff leafspace condition for leafwise quasi-geodesic foliations.
Researchers determine Baum-Bott residues for foliations without generic hypothesis.
problem Determining Baum-Bott residues for singular foliations without generic assumptions.
method Express residues in terms of Grothendieck residue and apply Cenkl's algorithm.
result Residues can be expressed in terms of a simpler foliation and algorithm holds without regularity assumption.
We obtain an asymptotic formula for the spectrum distribution function of the Laplace operator on a compact Riemannian Sol-manifold in the adiabatic limit determined by a one-dimensional foliation defined by the orbits of a left-invariant flow.
Characterizes flag geometries for Hitchin representations in SL3(R).
problem Understanding flag geometries associated with Hitchin representations in SL3(R).
method Geometric characterization based on invariant foliations and refraction flows.
result Constructs refraction flows for positive roots in general sl_n(R), with highest root flows being C^1+α.
Classifies Lie algebras actions on 3D spaces, focusing on solvable groups.
problem Classifying transitive actions of Lie algebras on 3D spaces.
method Local equivalence and structure of one-dimensional invariant foliations.
result Cannot extend classification to solvable case.
Study excess logarithmic residues for foliations to bound invariant hypersurfaces and test log canonicity.
problem Bounding invariant hypersurfaces and testing log canonicity of singularities.
method Introduce excess logarithmic residues, prove residue formula, derive Poincaré-type bound, and use them to recover log discrepancies.
result Componentwise logarithmic residues of a lifted foliation along the exceptional divisor recover log discrepancies of singularities.
In this article, we show that, for any compact 3-manifold, there is a C1 volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the exam…
Proves a Baum--Bott formula for foliations by curves with logarithmic terms.
problem Analyzing singularities and smoothness in foliations by curves.
method Logarithmic Baum--Bott residues for foliated triples (X,F,D), relating to Poincaré's Problem and GSV indices. result Logarithmic Baum--Bott residues generalize Aleksandrov logarithmic index for vector fields on hypersurfaces.
The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geo…
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.
Let D be a bounded logarithmically convex complete Reinhardt domain in Cn centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the C∗-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending …
In this paper we consider the question of bounding the degree of an divisor D invariant by a $\F$ holomorphic foliation, without rational first integral, on smooth algebraic variety X in terms of degree of $\F$ and some invariants of D and X. Particularly, if $\F$ is a foliation of degree d on $\mathbb{P}_{\m…
This paper connects foliations of the plane to non-Hausdorff 1-manifolds.
problem Connecting foliations of the plane to non-Hausdorff 1-manifolds.
method Establishes a connection between foliations of the plane and non-Hausdorff 1-dimensional manifolds.
result Establishes a beautiful connection between foliations of the plane and non-Hausdorff 1-dimensional manifolds.
Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.
problem Analyzing sub-Riemannian Brownian motions and their radial processes.
method Application of Itô's formula and sub-Laplacian comparison theorems to prove stochastic completeness and eigenvalue estimates.
result Proved Cheng's type estimates for Dirichlet eigenvalues of sub-Riemannian metric balls.
An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two such structures are locally diffeomorphic. We investigate the space of global deformations of canonical Engel structures arising out of contact three-manifolds. The main tool is Cartan's method of prolongation and d…
The paper presents a new approach to Sasakian manifolds using Riemannian foliations.
problem Understanding the geometric properties and structures of Sasakian manifolds.
method A foliated approach to Sasakian manifolds, focusing on cohomological properties.
result Formulation of obstructions to the existence of Sasakian structures on compact manifolds.
Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let θbe a contact form for (V,HT(V)), ξ_0 the Reeb vector field determined by θand assume that ξ_0 is of class C^k. In this paper we prove th…
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.
Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. Such flat surfaces are naturally organized into families which appear to b…
Study one-dimensional topological theories with linear generating functions.
problem Understanding one-dimensional topological theories with defects.
method Construct bases of hom spaces for decorated unoriented one-dimensional cobordisms.
result Gram determinant and linear generating functions constructed.
We found a new simple family of Cantor sets whose projections are one-dimensional.
problem Finding simple Cantor sets with specific projection properties.
method Developed a new series of self-similar Cantor sets in R3. result All projections of these new Cantor sets are connected and one-dimensional.
Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.
problem Understanding and characterizing one-dimensional non-Hausdorff manifolds.
method Analyzing properties of connected non-Hausdorff manifolds and their quotient spaces to CW complexes.
result Existence of a quotient map from a connected non-Hausdorff manifold to an open one-dimensional CW complex.
A new metric-based principal curve method learns 1D manifolds from spatial data.
problem Learning 1D manifolds from spatial data.
method Metric-based Principal Curve (MPC) approach.
result The method effectively learns the shape of 1D manifolds from synthetic and real datasets.
The Brasselet number helps calculate function germs with one-dimensional critical sets.
problem Calculating topological information of function germs with nonisolated singularities.
method Using the Brasselet number, the paper presents formulas for function germs with a one-dimensional critical locus.
result Formulas for function germs with a one-dimensional critical locus.
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 prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have t…
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.
One-dimensional crystals have convex shapes under certain conditions.
problem Determining if one-dimensional crystals have convex shapes.
method Analyzing the free energy under mass constraints and convexity assumptions.
result In one dimension, crystals have convex shapes under given 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.