Heat equation and ergodic theorems for Riemann surface laminations
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.
scientific article; zbMATH DE number 6084007
scientific article