Integral decompositions of varifolds

TL;DR

This paper develops a method to decompose integral varifolds into countably many parts, establishing existence under certain conditions and exploring generalizations, with implications for understanding varifold structure.

Contribution

It introduces a new decomposition framework for integral varifolds and proves existence results, extending to classes with bounded density.

Findings

Existence of decompositions for integral varifolds with integrable first variation

Generalization to rectifiable varifolds with bounded density

Decomposition may not be unique

Abstract

This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is established. Furthermore, this result can be generalized by replacing the class of integral varifolds by some classes of rectifiable varifolds whose density is uniformly bounded from below. However, such decomposition may fail to be unique.