# The concept of varifold

## TL;DR

This paper surveys the concept of varifold, a generalized notion of submanifold, focusing on regularity and variational properties of integral varifolds with mean curvature, essential for understanding complex geometric analysis problems.

## Contribution

It provides a minimal-prerequisite survey with 20 examples, emphasizing the regularity theory of integral varifolds and their role in geometric variational problems.

## Key findings

- Integral varifolds serve as a natural framework for variational surface theory.
- Regularity results for integral varifolds with mean curvature are discussed.
- Applications include existence and regularity of stationary and stable surfaces.

## Abstract

We survey - by means of 20 examples - the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one considers, for instance, existence or regularity of stationary (or, stable) surfaces of dimension at least three, or the limiting behaviour of sequences of smooth submanifolds under area and mean curvature bounds.