Metrics and convergence in the moduli spaces of maps
Joseph Palmer

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 and where is equipped with a volume form we consider families of maps in the collection and we define a distance function similar to the distance on such a collection. The definition of 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 we shift our focus to exploring the convergence properties of families of such maps.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory
