Polygonal approximations of closed parametric varifolds

TL;DR

This paper introduces holonomic measures as a new class of varifold analogues that encode local parameterization and orientation, and demonstrates their approximation by smooth chains with controlled properties, with applications to foliations.

Contribution

It defines holonomic measures for varifold analogues and proves their approximation by smooth chains with controlled boundaries and actions, enabling new applications.

Findings

Holonomic measures effectively encode local parameterization and orientation.

Smooth chain approximations with controlled properties are achievable for these measures.

Application demonstrated in the study of foliations on the torus.

Abstract

We define holonomic measures to be certain analogues of varifolds that keep track of the local parameterization and orientation of the submanifold they represent. They are Borel measures on the direct sum of several copies of the tangent bundle. We show that there is an approximation to these by smooth singular chains whose boundaries and Lagrangian actions are controlled. As an illustration of the usefulness of this result, we show how this can be applied to study foliations on the torus. We give other applications elsewhere.