arXiv research

Study shows conditions for continuity of foliated homeomorphisms action on space of leaves.

A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same con…

Study on the topology of leaves in singular Riemannian foliations.

Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.

Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.

In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.

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…

We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows fast…

In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering spacelike geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that monodromy of Margulis Space Times are "Anosov representations in non semi-simple Lie …

New proof of uniformization for hyperbolic foliations.

We construct smooth fiber bundles such that the fibers are exotic tori and the total space has finite abelian fundamental group. This gives examples of a Riemannian foliation on a closed manifold whose leaves are exotic tori and whose total space has finite abelian fundamental group.

The paper studies homeotopy groups of leaf spaces for specific foliations.

Proves conjecture about foliations on curved spaces.

Holomorphic foliations found in ball space with unique properties.

A singular riemannian foliation on a complete riemannian manifold is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immers…

Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $π_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}…

Molino's description of Riemannian foliations on compact manifolds is generalized to the setting of compact equicontinuous foliated spaces, in the case where the leaves are dense. In particular, a structural local group is associated to such a foliated space. As an application, we obtain a partial generalization of res…

We prove that (apart from dimension $n=4$), each Riemannian solenoidal lamination with transitive homeomorphism group and leaves isometric to a symmetric space $X$ of noncompact type, is homeomorphic to the inverse limit of the system of finite covers of a compact locally-symmetric $n$-manifold.

The paper explores symplectic foliations and their leaves on manifolds.

We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theor…

For a connected abelian Lie group T acting on a Poisson manifold (Y,π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves…

Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.

Thurston's boundary to the universal Teichmüller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of Teichmüller type if it shrinks vertical foliation of an integrable holomorphic quadratic differential. In a prio…

Let $X$ be an $(n+1)$-dimensional manifold, $Δ$ be a one-dimensional foliation on $X$, and $p: X \to X / Δ$ be a quotient map. We will say that a leaf $ω$ of $Δ$ is special whenever the space of leaves $X / Δ$ is not Hausdorff at $ω$. We present necessary and sufficient conditions for the map $p: X \to X / Δ$ to be a l…

We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in $C^\infty$ to…

It is proved that the isometry classes of pointed connected complete Riemannian $n$-manifolds form a Polish space, $\mathcal{M}_*^\infty(n)$, with the topology described by the $C^\infty$ convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifo…

Investigates dual foliations of polygon spaces based on area and perimeter.

Consider a parallel plane foliation on real finite-dimensional linear vector space. It induces a foliation on the torus obtained by factorization of the space by the integer lattice (let us denote the latter foliation by F). Let g be arbitrary metric on the torus. It induces a complex structure on each leaf of F such t…

Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $σ|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. We investigate situations where this foliation has compact leaves; in such cases we obtain a space of leaves Y/F which has dimension 2n-2 and admits…

Study wave invariants for Riemannian foliations, showing independence of mean curvature.

An equifocal submanifold M of a symmetric space N of compact type induces a foliation with singular leaves on N. In this paper we will show how to reconstruct the equifocal foliation starting from one of the singular leaves, the so-called focal manifolds. To be more concrete: The equifocal submanifold is equal to a par…

In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was given in \cite{drt} for Euclidean hypers…

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. Recently, we associated a new Riemannian metric to a codimension-one foliated Finsler space and proved integral formulae for general and for Randers spaces. In …

We develop a theory of "minimal $θ$-graphs" and characterize the behavior of limit laminations of such surfaces, including an understanding of their limit leaves and their curvature blow-up sets. We use this to prove that it is possible to realize families of catenoids in euclidean space as limit leaves of sequences of…

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a gener…

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…

A foliation on a manifold M can be informally thought of as a partition of M into injectively immersed submanifolds, called leaves. In this thesis we study foliations whose leaves carry some specific geometric structures. The thesis consists of two parts. In the first part we classify foliations on open manifolds whose…

Some properties of Riemannian foliations on closed manifolds are generalized to compact equicontinuous foliated spaces. For instance, it is proved that all holonomy covers of the leaves are quasi-isometric to each other.

For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of $L^2$ 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…

We show that which that for a Berwald structure, any Riemannian structure that is preserved by the Berwald connection leaves the indicatrix invariant under horizontal parallel transport. We also obtain the converse result: if $({\bf M},F)$ is a Finsler structure such that there exists a Riemannian structure that leaves…

We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to …

Totally geodesic dual leaves on curved manifolds are also curved.

Investigates singular Finsler foliations on $(α,β)$-spaces and their relation to Riemannian foliations.

We state some generalizations of a theorem due to G. Darboux, which originally states that a polynomial vector field in the complex plane exhibits a rational first integral and has all its orbits algebraic provided that it exhibits infinitely many algebraic orbits. In this paper, we give an interpretation of this resul…

Optimizes Lasso hyperparameters using leave-one-out CV.

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…

New examples of non-homeomorphic foliation leaves found.