Heat equation and ergodic theorems for Riemann surface laminations
type: Item
instance of: scholarly article
title: Heat equation and ergodic theorems for Riemann surface laminations
zbMATH Open document ID: 1331.37064
author: Tien-Cuong Dinh
author: Viêt Anh Nguyên
author: Nessim Sibony
published in: Mathematische Annalen
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.3931
review text: The goal of the paper under review is twofold: to prove ergodic theorems for possibly singular compact foliations by Riemann surfaces in complex manifolds, and to develop the study of the diffusion of the heat equation for laminations possibly with singularities with respect to a positive \(\partial\bar{\partial}\)-closed current tangent to the lamination. The theory of the heat diffusion with respect to a harmonic measure is developed, allowing the authors to extend the classical theory of \textit{A. Candel} and \textit{L. Conlon} [Foliations II. Graduate Studies in Mathematics 60. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1035.57001)]. This extention is done for laminations without any smoothness assumption on the transverse regularity. In particular, an ergodic theorem is obtained, as well as mixing results for the heat diffusion in a more general context. The authors study the heat equation with respect to a positive \(\partial\bar{\partial}\)-closed current \(T\) of bidimension \((1,1)\) with very weak regularity assumptions, and obtain ergodicity and mixing results for the heat equation with respect to a harmonic measure for singular foliations by Riemann surfaces in complex manifolds. An analogue of Birkhoff's ergodic theorem is given in the context of compact laminations with isolated singularities on complex manifolds.
reviewed by: Albetã Costa Mafra
Mathematics Subject Classification ID: 37F75
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 32C30
zbMATH DE Number: 6084007
zbMATH Keywords: harmonic currents
zbMATH Keywords: singular foliations
MaRDI profile type: Publication
OpenAlex ID: W1637441234
arXiv ID: 1004.3931
cites work: The \(\overline\partial\)-equation on a positive current
cites work: Inexistence of invariant measures for generic rational differential equations in the complex domain
cites work: Q4546485
cites work: Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices. (Genericity of non-zero Lyapunov exponents for deterministic products of matrices)
cites work: Uniformization of surface laminations
cites work: The harmonic measures of Lucy Garnett
cites work: Q4421930
cites work: Uniformization of the leaves of a rational vector field
cites work: Q3344779
cites work: Q5571891
cites work: Q5181909
cites work: Q4039882
cites work: Q5565773
cites work: Complex Hénon mappings in \({\mathbb{C}}^ 2\) and Fatou-Bieberbach domains
cites work: Harmonic currents of finite energy and laminations
cites work: Riemann surface laminations with singularities
cites work: Unique ergodicity of harmonic currents on singular foliations of \({\mathbb{P}^2}\)
cites work: Examples of minimal laminations and associated currents
cites work: Foliations, the ergodic theorem and Brownian motion
cites work: Q5434578
cites work: Equations de Pfaff algébriques
cites work: Brownian motion on foliations: Entropy, invariant measures, mixing
cites work: Q4840149
cites work: Uniformization and the Poincar� metric on the leaves of a foliation by curves
cites work: Algebraic solutions of one-dimensional foliations
cites work: An Operator Theorem on $L_1$ Convergence to Zero with Applications to Markov Kernels
cites work: Prolongement des courants, positifs, fermes de masse finie
cites work: ON THE MAXIMAL ERGODIC THEOREM
cites work: Cycles for the dynamical study of foliated manifolds and complex manifolds
cites work: Q4169288
cites work: Q3160334
DOI: 10.1007/S00208-011-0730-8
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description: scientific article; zbMATH DE number 6084007