Metrics and convergence in the moduli spaces of maps

TL;DR

This paper develops a general framework for analyzing convergence of families of measurable maps between manifolds, introducing a parameter-independent metric that ensures completeness and facilitates the study of convergence properties.

Contribution

It introduces a new metric framework for measurable maps between manifolds that is independent of parameter choices and proves the resulting space is complete.

Findings

The metric space of measurable maps is complete.

The topology of the space does not depend on parameter choices.

The framework applies to convergence analysis of families of maps.

Abstract

We provide a general framework to study convergence properties of families of maps. For manifolds $M$ and $N$ where $M$ is equipped with a volume form $\mathcal{V}$ we consider families of maps in the collection $\{(\phi, B) : B \subset M, \phi:B \rightarrow N\text{ with both measurable}\}$ and we define a distance function $\mathcal{D}$ similar to the $L^1$ distance on such a collection. The definition of $\mathcal{D}$ depends on several parameters, but we show that the properties and topology of the metric space do not depend on these choices. In particular we show that the metric space is always complete. After exploring the properties of $\mathcal{D}$ we shift our focus to exploring the convergence properties of families of such maps.