Approximability for Lagrangian submanifolds

TL;DR

This paper introduces a new concept of categorical approximability for metric spaces, extending prior notions, and demonstrates its applicability to various classes of Lagrangian submanifolds, revealing new geometric properties.

Contribution

It defines a novel categorification of approximability for metric spaces and applies it to Lagrangian submanifolds, showing their approximability beyond precompactness.

Findings

Several classes of Lagrangian submanifolds are shown to be approximable.

Examples of approximable Lagrangian spaces that are not precompact.

The concept generalizes previous notions of approximability in geometry.

Abstract

This paper introduces a notion of categorical approximability for metric spaces that can be viewed as a categorification of approximability for metric groups, as defined by Turing in 1938. Approximability as introduced here is a property of metric spaces that is more general than precompactness. It is shown that several classes of Lagrangian submanifolds - closed Lagrangian submanifolds in a cotangent disk bundle; equators on the sphere; weakly exact Lagrangians on the torus-endowed with the spectral metric are approximable in this sense. Among other geometric applications, we show that there are such examples of spaces of Lagrangians that are approximable but are not precompact.