A Compactness Theorem for Embedded Measured Riemann Surface Laminations

TL;DR

This paper establishes a compactness theorem for embedded measured hyperbolic Riemann surface laminations within compact almost complex manifolds, utilizing a Levy-Prokhorov metric topology, and applies it to biholomorphisms fixing laminations.

Contribution

It introduces a new compactness theorem for measured Riemann surface laminations and demonstrates its application to biholomorphisms in complex manifolds.

Findings

Proved a compactness theorem for measured Riemann surface laminations.

Established a topology on the space of laminations using Levy-Prokhorov metric.

Showed some power of a biholomorphism fixes a lamination.

Abstract

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of the compactness theorem, we show that given a biholomorphism of $\phi $ of a closed complex manifold $X$, some power $\phi^k $ ($k>0$) fixes a measured Riemann surface lamination in $X$.