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.
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48 results for “Morse foliations”
Study Morse-Novikov cohomology on foliated manifolds and prove Hodge theorem.
problem Understanding cohomology groups on foliated manifolds.
method Defined and studied Morse-Novikov cohomology relative to a foliation, proving homotopy invariance and extending to more general forms.
result Proved Hodge theorem and Poincaré duality for reduced leafwise Morse-Novikov cohomology groups on Riemannian foliations.
The paper studies Morse theory for Lie algebra actions on Riemannian foliations.
problem Morse theory for Lie algebra actions on Riemannian foliations.
method Equivariant Morse-Bott theory on leaf space.
result Established foliated versions of Morse-Bott lemma and handle presentation theorem.
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.
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.
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.
The paper extends Witten's deformation to foliations and Morse functions.
problem Extending Witten's deformation to foliations and Morse functions.
method Using deformation to the normal cone and C*-modules, the paper constructs the Witten deformation for generic functions on foliations.
result Establishes the compactness of the resolvent and Morse inequalities for foliations with invariant transverse measures.
We study codimension one smooth foliations with Morse type singularities on closed ma-nifolds. We obtain a description of the manifold in case the number of centers in greater then the number of saddles. This result relies on and extends previous results of Reeb (for foliations having only centers) and Ells-Kuiper (for…
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.
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.
Researchers simplify the computation of diffeomorphism groups for Morse-Bott foliations.
problem Computing the homotopy type of diffeomorphism groups for Morse-Bott foliations.
method Reduces the computation to three groups: diffeomorphisms of the critical manifold, vector bundle automorphisms, and fixed near the critical manifold.
result Shows how to compute the homotopy type of diffeomorphism groups for certain Morse-Bott foliations.
Study Teichmüller dynamics and dilation tori properties, proving almost all vertical foliations are Morse-Smale.
problem Understanding the coarse geometry of dilation tori and Teichmüller flow dynamics.
method Analysis of moduli space of dilation tori, Teichmüller flow action, and piecewise affine circle homeomorphisms.
result Vertical foliations of dilation tori are almost always Morse-Smale.
Generalizes Giroux's result to higher dimensions and applies to contact manifolds.
problem Characterizing convex surfaces in contact manifolds.
method Extending Giroux's result to arbitrary dimensions and applying to specific hypersurfaces.
result Closed hypersurfaces are -close to convex hypersurfaces.
Study topological properties of foliations induced by closed 1-forms on orbifolds.
problem Characterize the topology of foliation leaves induced by closed 1-forms on orbifolds.
method Establish criteria for the compactness of foliation leaves and extend a topological result to orbifolds.
result Criteria for the compactness and coexistence of foliation leaves are established.
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.
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
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.
In this paper we find sufficient conditions for the vanishing of the Morse-Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian…
New Teichmüller geodesic rays found with unique foliations.
problem Capturing generic directions in Teichmüller space.
method Using Chaika-Masur-Wolf and Durham-Zalloum work.
result First sublinearly-Morse geodesic rays with minimal non-uniquely ergodic foliations.
We study codimension one (transversally oriented) foliations $\fa$ on oriented closed manifolds 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…
In case of the heat flow on the free loop space of a closed Riemannian manifold non-triviality of Morse homology for semi-flows is established by constructing a natural isomorphism to singular homology of the loop space. The construction is also new in finite dimensions. The main idea is to build a Morse filtration usi…
Paper presents local versions of Demailly-Bouche's holomorphic Morse inequalities.
problem Asymptotic bounds for cohomology groups on Hermitian manifolds.
method Variation of Berman's method for compact complex manifolds with boundary.
result Local versions of holomorphic Morse inequalities hold on any Hermitian manifold.
This paper presents a natural extension to foliated spaces of the following result due to Gromov : the h-principle for open, invariant differential relations is valid on open manifolds. The definition of openness for foliated spaces adopted here involves a certain type of Morse functions. Consequences concerning the pr…
Develops graphical calculus for monoidal categories with twisted pivotal structures.
problem Constructing modules for surfaces with Morse functions or foliations.
method Graphical calculus and string nets for monoidal categories with twisted pivotal structures.
result Twisted string net modules assemble in an oriented categorified 2-TQFT.
We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian
Let be a smooth manifold and let $\F$ be a codimension one, foliation on , with isolated singularities of Morse type. The study and classification of pairs $(M,\F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb \cite{Reeb} states that a manifold admitting a …
Foliated Open Booksmath.GT
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.
We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a prope…
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…
We summarize the foliation approach to compactifications of eleven-dimensional supergravity on eight-manifolds down to spaces for the case when the internal part of the supersymmetry generator is chiral on some proper subset of . In this case, a topological no-go theo…
Zeta invariants study Morse forms on Riemannian manifolds, proving smoothness and convergence.
problem Analyzing zeta invariants of Morse forms on Riemannian manifolds.
method Perturbations of de~Rham derivatives and Laplacians, heat semigroup, instantons, Mathai-Quillen currents.
result ζ(1,z) converges to a real number z as μ → ±∞ for Morse forms, describing preserved leaves in foliated flows.
The basic cohomology of a Riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing fo…
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
We prove diffeomorphisms of polygonal linkage moduli spaces to Euclidean spaces.
problem Moduli spaces of self-avoiding polygonal linkages and configurations.
method Construct Lyapunov-Reeb functions to show diffeomorphisms.
result Moduli spaces are diffeomorphic to Euclidean spaces.
We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple. Furtherm…
A surface diffeo reversing orientation is isotopic to identity.
problem Understanding diffeomorphisms reversing orientation on surfaces.
method Analyzing Morse functions and diffeomorphisms preserving them.
result Reversing orientation homeomorphisms squared are isotopic to identity.
We use the theory of singular foliations to study compactifications of eleven-dimensional supergravity on eight-manifolds down to spaces, allowing for the possibility that the internal part of the supersymmetry generator is chiral on some locus which does not coincide wi…
Floer theory connects dynamics on surfaces to their chain-level theory.
problem Connecting dynamics on surfaces to their Floer theory.
method Using capped 1-periodic orbits and ideas from Hofer-Wysocki-Zehnder's theory.
result Definition and computation of novel spectral invariants.
Constructs uniformly positive scalar curvature metrics on open manifolds
problem Finding uniformly positive scalar curvature metrics on open manifolds
method Using Morse functions and exhaustion
result Proving the existence of uniformly positive scalar curvature metrics
A new distance measure for circular Heegaard splittings helps understand knot exteriors.
problem Understanding the structure of knot exteriors using circular Heegaard splittings.
method Defining and analyzing circular distance for circular Heegaard splittings.
result Circular distance bounds properties of knot exteriors, like the uniqueness of minimal-genus Seifert surfaces.
We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to new proofs of certain results previously obtained using Seiberg-Witten monopole Floer hom…
New method extends discrete Morse theory to simplicial complexes.
problem Discrete Morse theory on simplicial complexes.
method Morse shellings and compatible discrete Morse functions.
result Triangulated surfaces and manifolds have Morse shellable triangulations.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
Discrete Morse-Bott theory on CW complexes generalizes Forman's theory.
problem No specific problem stated; focuses on theory development.
method Derived a discrete Morse-Bott theory on CW complexes.
result Discrete Morse-Bott theory is a generalization of Forman's theory.
New proof for discrete Morse theory using combinatorial construction.
problem Verifying the Morse differential in discrete Morse homology.
method Combinatorial construction of flowlines in discrete Morse theory.
result Morse differential squares to zero in discrete Morse homology.
Characterizes geodesics on spheres with Morse index bounds and inequalities.
problem Understanding geodesics on spheres using Morse theory.
method Morse-theoretic characterization and strong Morse inequalities.
result Existence of geodesics with specific Morse indices on spheres.
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
Morse-Conley-Floer Homologymath.DS
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…