arXiv research

Paper introduces entropy for Riemannian foliations and connects it to Ricci flow.

Defines and calculates foliation homology from flows.

The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.

Study of foliations on symmetric spaces and mean curvature flow results.

We describe transversely oriented foliations of codimension one on closed manifolds that admit simple foliated flows.

Study on minimal foliations in 3D manifolds with specific conditions.

In this paper, we investigate the mean curvature flows starting from all non-minimal leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infin…

A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered…

The paper studies Liouville structures for taut foliations and Anosov flows, proving their topological invariance.

We prove a general result about the short time existence and uniqueness of second order geometric flows transverse to a Riemannian foliation on a compact manifold. Our result includes some flows already existing in literature, as the transverse Ricci flow, the Sasaki-Ricci flow and the Sasaki J-flow and motivates the s…

Extends Masur's divergence theorem to complex tori and Kummer surfaces.

Study on transverse Ricci solitons on compact foliated manifolds.

We prove the existence of foliations transverse to pseudo-Anosov flows using veering triangulations.

The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $β$ defining a foliation on a branched simple polyhedron $P$ satisfies $dβ>0$, which…

Paper constructs flows converging to cones and foliations.

Transverse one dimensional foliations play an important role in the study of codimension one foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one foliation and that of a give…

The paper explores universal circles for Anosov foliations and their uniqueness.

Study of centralizer elements preserving geodesic flow foliations on covers.

The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.

Reeb flow made transverse to foliations without invariant measures.

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of t…

Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.

Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of…

Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliatio…

In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, C. -Y. Lin used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension and C. Sormani and Lin proved useful results with this …

New proof shows left orderability of 3-manifold groups with specific foliations.

We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …

Characterizes flag geometries for Hitchin representations in SL3(R).

Study shows how certain foliations in unit tangent bundles behave.

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…

A flow of metrics, $g_t$, 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…

The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.

We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the…

Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.

Study shows universal circle isomorphic to flow space ideal boundary.

We prove here that given a proper isometric action $K\times M\to M$ on a complete Riemannian manifold $M$ then every continuous isometric flow on the orbit space $M/K$ is smooth, i.e., it is the projection of an $K$-equivariant smooth flow on the manifold $M$. As a direct corollary we infer the smoothness of isometric …

New 3D shapes found without certain flows.

Anosov flow found in specific partially hyperbolic systems.

Study mean curvature flow in hyperbolic 3-manifolds, proving foliations and existence of minimal surfaces.

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

Paper defines dynamical coherence for flows and proves it under specific conditions.

Completes preliminary structures in 3D flows to foliations.

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…

Let $M$ be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on $M$ to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes c…

Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…

Paper shows how to transform certain flows into R-covered ones.