Heat equation and ergodic theorems for Riemann surface laminations

TL;DR

This paper develops heat equation techniques for Riemann surface laminations, proving ergodic theorems related to heat diffusion and Birkhoff averaging, with applications to holomorphic foliations and harmonic measures.

Contribution

It introduces a heat equation framework for invariant currents in Riemann surface laminations, extending ergodic theorems to singular and real lamination contexts.

Findings

Proves ergodic theorems for heat diffusion on Riemann surface laminations.

Establishes Birkhoff-type averaging results for invariant currents.

Develops heat diffusion theory for harmonic measures in real laminations.

Abstract

We introduce the heat equation relative to a positive dd-bar-closed current and apply it to the invariant currents associated with Riemann surface laminations possibly with singularities. The main examples are holomorphic foliations by Riemann surfaces in projective spaces. We prove two kinds of ergodic theorems for such currents: one associated to the heat diffusion and one close to Birkhoff's averaging on orbits of a dynamical system. The heat diffusion theorem with respect to a harmonic measure is also developed for real laminations.