Equality of the usual definitions of Brakke flow

TL;DR

This paper demonstrates the equivalence of various definitions of Brakke flow, a concept in geometric measure theory, by correcting a key proof in Brakke's original work.

Contribution

It establishes the equivalence of multiple variants of Brakke flow definitions and corrects a crucial proof in the foundational literature.

Findings

Most definitions of Brakke flow are equivalent.

A correction is provided for Brakke's original estimate in .5.

Clarifies foundational aspects of mean curvature flow in geometric measure theory.

Abstract

In 1978 Brakke introduced the mean curvature flow in the setting of geometric measure theory. There exist multiple variants of the original definition. Here we prove that most of them are indeed equal. One central point is to correct the proof of Brakke's \S 3.5, where he develops an estimate for the evolution of the measure of time-dependent test functions.