Harmonic measures in embedded foliated manifolds (original) (raw)

Foliated stochastic calculus: Harmonic measures

Transactions of the American Mathematical Society, 2015

In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.

Degenerate Semigroups and Stochastic Flows of Mappings in Foliated Manifolds

Potential Analysis, 2015

Let (M, F) be a compact Riemannian foliated manifold. We consider a family of compatible Feller semigroups in C(M n) associated to laws of the n-point motion. Under some assumptions (Le Jan and Raimond, [16]) there exists a stochastic flow of measurable mappings in M. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated.

An averaging principle for diffusions in foliated spaces

Consider an SDE on a foliated manifold whose trajectories lay on compact leaves. We investigate the effective behavior of a small transversal perturbation of order ε. An average principle is shown to hold such that the component transversal to the leaves converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to invariant measures on the leaves, as ε goes to zero. An estimate of the rate of convergence is given. These results generalize the geometrical scope of previous approaches, including completely integrable stochastic Hamiltonian system.

Statistical behaviour of the leaves of Riccati foliations

Ergodic Theory and Dynamical Systems, 2010

We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension 1 and hyperbolic, corresponding to the unique complete metric of curvature -1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivisation of the solutions of a linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ : π 1 (S) → GL(n, C). We give conditions under which the foliated geodesic flow has a generic repellor-attractor statistical dynamics. That is, there are measures µ − and µ + such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to µ − and for positive time to µ + (i.e. µ + is the unique SRB-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.

Unique Ergodicity of Harmonic Currents On Singular Foliations of [FORMULA]

Geometric and Functional Analysis, 2010

Let F be a holomorphic foliation of P 2 by Riemann surfaces. Assume all the singular points of F are hyperbolic. If F has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F . This implies strong ergodic properties for the foliation F . We also study the harmonic flow associated to the current T.

and R.Wolak: Deforming metrics of foliations

2016

We study geometry of a manifold endowed with two complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the metric varies conformally along one of the distributions. Then we introduce the geometric flow depending on the mean curvature vector of the distribution, and show existence/uniquenes and convergence of a solution as t → ∞, when the complementary distribution is integrable with compact leaves. We apply the method to the problem of prescribing mean curvature vector field of a foliation, and give examples for harmonic and umbilical foliations and for the double-twisted product metrics, including the codimension-one case.

Extrinsic geometric flows on foliated manifolds, III

arXiv: Differential Geometry, 2010

AbstractThe geometry of a codimension-one foliation with a time-dependent Riemannian metric isstudied. The work begins with formulae concerning deformation of geometric quantities as theRiemannian metric varies along the leaves of the foliaiton. Then the Extrinsic Geometric Flowdepending on the secondfundamental formofthe foliation isintroduced. Under suitableassump-tions, this evolution yields the second order parabolic PDEs, for which the existence/uniquenesand in some cases converging of a solution are shown. Applications to the problem of prescrib-ing mean curvature function of a codimension-one foliation, and examples with harmonic andumbilical foliations (e.g., foliated surfaces) and with twisted product metrics are given. 1 Introduction A Geometric Flow (GF) is an evolution of a given geometric structure under a differential equationrelated to a functional on a manifold, usually associated with some curvature. The most popularGFs in mathematics are the Heat flow, the Ricci flow ...

UNIQUE ERGODICITY FOR FOLIATIONS IN P 2 WITH AN INVARIANT CURVE

Consider a foliation in the projective plane admitting a projective line as the unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a unique positive dd c-closed (1, 1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. The result uses an extension of our theory of densities for currents. Foliations on compact Kähler surfaces are also considered.

Deforming metrics of foliations

Central European Journal of Mathematics, 2013

Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.

On mean curvature flow of singular Riemannian foliations: Noncompact cases

Differential Geometry and its Applications, 2020

In honor of Professor Jürgen Berndt's 60th birthday. [ACG19] Marcos M. Alexandrino, Leonardo F. Cavenaghi and Icaro Gonçalves, Mean curvature flow of singular Riemannian foliations: Non compact cases, arXiv:1909.04201 (2019) On Mean curvature flow of Singular Riemannian foliations: Non compact cases Marcos M. Alexandrino (IME-USP) In honor of Professor Jürgen Berndt's 60th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1 Definition Given a Riemannian manifold M and an immersion ϕ : L 0 → M, a smooth family of immersions ϕ t : L 0 → M, t ∈ [0, T) is called a solution of the mean curvature flow (MCF for short) if ϕ t satisfies the evolution equation d dt ϕ t (x) = H(t, x), where H(t, x) is the mean curvature of L(t) := ϕ t (L 0). On Mean curvature flow of Singular Riemannian foliations: Non compact cases Marcos M. Alexandrino (IME-USP) In honor of Professor Jürgen Berndt's 60th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1 Definition A submanifold L of a space form M(k) is called isoparametric if its normal bundle is flat and the principal curvatures along any parallel normal vector field are constant. On Mean curvature flow of Singular Riemannian foliations: Non compact cases Marcos M. Alexandrino (IME-USP) In honor of Professor Jürgen Berndt's 60th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1 Definition A singular foliation F = {L} is called a generalized isoparametric if 1 F is Riemannian, i.e., every geodesic perpendicular to one leaf is perpendicular to every leaf it meets. On Mean curvature flow of Singular Riemannian foliations: Non compact cases Marcos M. Alexandrino (IME-USP) In honor of Professor Jürgen Berndt's 60th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1 Definition A singular foliation F = {L} is called a generalized isoparametric if On Mean curvature flow of Singular Riemannian foliations: Non compact cases Marcos M. Alexandrino (IME-USP) In honor of Professor Jürgen Berndt's 60th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1 Definition A singular foliation F = {L} is called a generalized isoparametric if On Mean curvature flow of Singular Riemannian foliations: Non compact cases Marcos M. Alexandrino (IME-USP) In honor of Professor Jürgen Berndt's 60th birthday.