Statistical behaviour of the leaves of Riccati foliations (original) (raw)

The Foliated Geodesic Flow on Riccati Equations (Complex Dynamics)

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

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.

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 ...

Harmonic measures in embedded foliated manifolds

Stochastics and Dynamics, 2016

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold. We construct explicitly a Stratonovich differential equation for the foliated Brownian motion. We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.

The partial Ricci flow on one-dimensional foliations

2013

We first consider a one-dimensional foliation, since this case is easier. We prove local existence/uniqueness theorem, deduce the government equations for the curvature and conullity tensors (which are parabolic along the leaves), and show convergence of solution metrics for some classes of almost-product structures. For the warped product initial metric the global solution metrics converge to one with constant mixed sectional curvature.

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.

Uniformization of compact foliated spaces by surfaces of hyperbolic type via the Ricci flow

Proceedings of the American Mathematical Society, 2022

We give a new proof of the uniformization theorem of the leaves of a compact lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversally continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.

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.

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.