Some Applications of the Isoperimetric Inequality for Integral Varifolds

TL;DR

This paper leverages the isoperimetric inequality for integral varifolds to derive precise estimates on the measure of regions with low density quotient and extends classical differentiability theories to the setting of integral varifolds.

Contribution

It introduces new sharp estimates for integral varifolds and generalizes foundational differentiability results from Lebesgue measure to varifolds.

Findings

Sharp estimates for the measure of low-density regions.

Extension of Calderón-Zygmund differentiability theory to varifolds.

Broader applicability of isoperimetric inequalities in geometric measure theory.

Abstract

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calder\'on's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.