Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and ( F , F ′ ) f (F,F')_f ( F , F ′ ) f -harmonic maps. result Equivalence of transversally f-harmonic and ( F , F ′ ) f (F,F')_f ( F , F ′ ) f -harmonic maps in minimal foliations. 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.
The paper studies critical points of horizontal energy functional in Riemannian foliations.
problem Analyzing critical points of horizontal energy functional in Riemannian foliations.
method Utilizing stress-energy tensor, establishing monotonicity formulas, and Jin-type theorems.
result Established monotonicity formulas for horizontally harmonic maps and transversally harmonic maps.
Study foliations from complex ball to another via harmonic maps.
problem Rigidity of complex ball quotients.
method Lattice-equivariant harmonic map of small rank.
result Rigidity of complex ball quotients proven.
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.
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.
Paper finds lower bounds for foliation maps under spin condition.
problem Finding bounds for foliation maps.
method Applied Llarull's theorem to foliations.
result Determined lower bounds for Lipschitz constants.
Proves generalized Chen's conjecture for biharmonic maps on foliations.
problem Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
method Analyzes (F,F')-biharmonic maps and their critical points.
result Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
Examines how meridians and parallels help in map drawing.
problem Understanding map drawing and foliations of the sphere.
method Analyzes Euler's work on cartography and meridians/parallels.
result Meridians and parallels are crucial for map drawing.
Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.
problem Computing stretch factors and foliations for pseudo-Anosov mapping classes efficiently.
method Quadratic-time algorithm using input word and length as complexity measure.
result First algorithm to compute stretch factors and foliations in sub-exponential time.
Taut foliations map leaves to branched 2-sphere covers.
problem Understanding taut foliations in 3-manifolds.
method Map foliation leaves to branched 2-sphere covers.
result Taut foliations are characterized by such maps.
A singular foliation on a complete riemannian manifold M is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This …
The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.
problem Analyzing maps between pseudo-Hermitian and Kähler manifolds.
method Investigates partial energy functionals and critical maps, proving foliated results for ∂ ‾ b \overline{\partial}_{b} ∂ b - and ∂ b \partial_{b} ∂ b -harmonic maps. result Generalizes Siu's holomorphicity result to ∂ ‾ b \overline{\partial}_{b} ∂ b - and ∂ b \partial_{b} ∂ b -harmonic maps. We study the transversally harmonic maps between foliated Riemannian manifolds. In particular, we prove that under some curvature conditions, any transversally harmonic map is transversally totally geodesic.
Anosov maps study with new Banach space and foliation method.
problem Understanding statistical properties of Anosov maps.
method Constructing a new Banach space and using a new foliation method.
result New Banach space provides insights into foliation absolute continuity.
Study the pullbacks and blowups of Lie algebroids and related structures.
problem Understanding the relationship between Lie algebroids, singular foliations, and Dirac structures under maps.
method Examine pullbacks and blowups of Lie algebroids and related structures under maps with constant rank or transversality assumptions.
result Establish the relation between the blowup of a Lie algebroid and its singular foliation.
The paper proves a Liouville theorem for a specific type of harmonic maps on foliated manifolds.
problem Investigating harmonic maps on foliated Riemannian manifolds.
method First variational formulas, generalized Weitzenböck type formula, and Liouville type theorem for ( F , F ′ ) p (\mathcal F,\mathcal F')_{p} ( F , F ′ ) p -harmonic maps. result Established a Liouville type theorem for ( F , F ′ ) p (\mathcal F,\mathcal F')_{p} ( F , F ′ ) p -harmonic maps. New maps connect universal circles to ideal sphere for hyperbolic manifolds.
problem Understanding universal circles for Anosov foliations with branching.
method Introduced a new type of Cannon--Thurston map for leftmost universal circles.
result Fundamental group acts on leftmost universal circle with pseudo-Anosov dynamics.
The author studies regions foliated by 1D families of functions and their applications.
problem Understanding regions represented as foliated forms and natural smooth maps onto them.
method Investigates natural smooth maps respecting canonical projections and moment maps, focusing on foliated regions.
result Discusses the 1st derivative of functions and critical sets in foliated regions.
Proves an index theorem for foliations using spectral triples.
problem Proving an Atiyah L 2 L^2 L 2 covering index theorem for foliations. method Symbol calculus for foliations and spectral triples.
result Induces the same map on K-theory for two types of spectral triples.
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.
The volume of a k-dimensional foliation F \mathcal{F} F in a Riemannian manifold M n M^{n} M n is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a construction by Gluck and Ziller, "singular" foliations by 3-spheres are constructed on…
Abstract shows mapping between foliation characteristic classes.
problem Mapping characteristic classes of foliations.
method Losik's approach to Gelfand formal geometry and Crainic-Moerdijk's Čech-de~Rham cohomology.
result Map between characteristic classes is non-injective.
Study describes algebraic structure of foliated homeomorphisms.
problem Algebraic structure of foliated homeomorphisms.
method Analysis of homeotopy groups of tree-like foliations.
result Description of algebraic structure of π0H+.
The paper studies critical maps of a specific energy functional on pseudo-Hermitian manifolds.
problem Investigating critical maps of a horizontal energy functional on pseudo-Hermitian manifolds.
method Deriving a CR Bochner formula and introducing a Paneitz type operator to refine the Bochner formula.
result Established Bochner type theorems and Lichnerowicz type results for ( H , H ~ ) (H,\widetilde{H}) ( H , H ) -harmonic maps. Study on Clairaut maps from nearly Kahler to Riemannian manifolds.
problem Characterizing Clairaut maps from nearly Kahler manifolds.
method Analyzing conditions for Clairaut maps to be totally geodesic foliations.
result Non-trivial examples of Clairaut maps are provided.
In this paper we study singular riemannian foliations that have sections,i.e., totally geodesic complete immersed submanifolds that meet each leaf orthogonally and whose dimensions are the codimensions of the regular leaves. We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an…
Develops Chern-Weil theory for singular foliations.
problem Chern-Weil theory for Haefliger-singular foliations.
method Constructs explicit forms representing characteristic classes in de Rham cohomology.
result Theory applies to general smooth Haefliger structures up to homotopy.
The study describes how quadratic differentials influence foliations on Riemann surfaces.
problem Understanding how quadratic differentials affect foliations on Riemann surfaces.
method Analyzing infinite-energy harmonic maps from Riemann surfaces to R-trees with prescribed behavior at poles.
result Any measured foliation is uniquely realized by a meromorphic quadratic differential with prescribed principal parts at poles.
The paper generalizes a theorem and introduces a new characteristic map for foliated manifolds.
problem The challenge is to generalize Bott's vanishing theorem for foliated manifolds.
method The approach involves working with the full holonomy groupoid instead of the Morita equivalent étale groupoid, leading to novel geometric representatives of characteristic classes.
result A characteristic map encoding both primary and secondary characteristic classes is introduced.
Sharp spectral estimates for negatively curved foliations.
problem Estimating the bottom of the spectrum of Riemannian foliations.
method Analyzing the normal exponential map and using it to derive spectral estimates.
result Sharp estimates for the bottom of the spectrum of Riemannian foliations.
The paper explores how geometric structures on orbifolds relate to foliations and applies this to harmonic maps.
problem Understanding geometric structures on orbifolds and their applications.
method Translation of classical geometrical structures to orbifolds and foliated approach to harmonic maps.
result The classical theory of geometrical structures translates to orbifolds and is related to foliated geometrical structures.
Consider a singular Riemannian foliation (s.r.f for short) on a compact manifold. By successive blow-ups along the strata, we construct a regular Riemannian foliation on another compact Riemannian manifold and a desingularization map that projects leaves of the regular Riemannian foliation into leaves of the s.r.f. Thi…
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.
The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.
problem Left-orderability of fundamental groups in Dehn fillings of pseudo-Anosov mapping tori.
method Two approaches: one using R \mathbb{R} R -covered foliations and the other using one-sided branching. result All such Dehn fillings have left-orderable fundamental groups.
Maps on foliated manifolds decrease area and scalar curvature is negative.
problem Understanding scalar curvature and area decreasing maps on foliated manifolds.
method Analyzing the scalar curvature and using properties of area decreasing maps.
result Negative scalar curvature on the support of the differential of the map.
We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a c…
The paper proves isomorphisms between two complexes related to singular foliations.
problem Understanding the isomorphisms between two complexes associated with singular foliations.
method Analyzing the quotient map and proving isomorphisms in specific cases.
result Isomorphisms between the complexes of differential forms on the leaf space and basic differential forms on the manifold.
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes wil…
Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
A singular Riemannian foliation F F F on a complete Riemannian manifold M M M is called a polar foliation if, for each regular point p p p , there is an immersed submanifold Σ Σ Σ , called section, that passes through p p p and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the part…
The paper proves properties of Finsler submanifolds and analytic maps.
problem Analyzing properties of Finsler submanifolds and their analytic maps.
method Proving properties of regular fibers of analytic maps and Finsler submersions.
result Regular fibers of an analytic map are equifocal under certain conditions.
Study of Matsumoto maps on foliated bundles over hyperbolic manifolds.
problem Characterizing ergodic harmonic measures on foliated bundles.
method Analysis of actions of hyperbolic manifold groups on the circle.
result Suspension of actions with non-discrete images cannot admit Matsumoto maps of type I.
Study on mapping class groups of non-orientable surfaces, proving some conjectures and refuting others.
problem Analogies between Fuchsian groups and mapping class groups of non-orientable surfaces.
method Analyzing limit sets, foliations, and geometric properties.
result Established parts of a conjecture about the limit set and provided evidence for and against the analogy.
This paper simplifies complex nonholonomic systems using momentum map reduction.
problem Reducing complex nonholonomic systems with symmetries.
method Using nonholonomic momentum bundle map and gauge transformation.
result Reduced manifolds are Chaplygin-type leaves with an almost symplectic form.
This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thursto…
Paper translates train track concepts to cluster algebras for pseudo-Anosov mapping classes.
problem Understanding pseudo-Anosov mapping classes on surfaces.
method Using Goncharov--Shen's potential function, the paper translates train track concepts into cluster algebra language.
result Proves sign stability of general pseudo-Anosov mapping classes.