Optimal smooth approximation of integral cycles

TL;DR

This paper proves that integral cycles in oriented Riemannian manifolds can be smoothly approximated within their homology class, maintaining nearly the same area and controlling singularities, advancing geometric measure theory.

Contribution

It introduces a method to approximate integral cycles by smooth submanifolds with minimal area difference and singularity control, improving understanding of cycle regularity.

Findings

Approximation of integral cycles by smooth submanifolds with nearly the same area.

Singular set of the approximation has codimension at least 5.

If homology class is smooth, the approximation is smooth without singularities.

Abstract

In this article we prove that each integral cycle $T$ in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by an integral cycle in the same homology class which is a smooth submanifold $\Sigma$ of nearly the same area, up to a singular set of codimension 5. Moreover, if the homology class $\tau$ is representable by a smooth submanifold, then $\Sigma$ can be chosen free of singularities.