Compactness theorems for the spaces of distance measure spaces and Riemann surface laminations

TL;DR

This paper extends Gromov's compactness theorem to the space of distance measure spaces and Riemann surface laminations, establishing new compactness results and linking them to the Deligne-Mumford compactification.

Contribution

It generalizes Gromov's compactness theorem using a new distance metric and connects the moduli space of Riemann surfaces to this framework.

Findings

Deligne-Mumford compactification is the completion of the moduli space under the new distance.

Established a compactness theorem for Riemann surface laminations.

Extended Gromov's theorem to a broader class of metric measure spaces.

Abstract

In this paper, we give a generalisation of Gromov's compactness theorem for metric spaces, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with a \emph{generalised Gromov-Hausdorff-Levi-Prokhorov distance}. Using this result we prove that the Deligne-Mumford compactification is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorff-Levi-Prokhorov distance. Further we prove a compactness theorem for the space of Riemann surface laminations.