Background for Varifold theory

Question

I noticed this question posted on MO, hence I estimated that this may be an acceptable question even in MO (and not for MSE). I studied the notion of current and in a nutshell I understood "varifolds are weaker objects than currents."

My question is what kind of prerequisites one needs to have in order to study varifold theory and varifold geometry. For example, as far as currents are concerned, one needs to have grasped the notions of geometric measure theory and a bit of multilinear algebra and the introduction to distribution theory in functional analysis. I would like to ask the principal prerequisites one needs to know to fully grasp varifold theory. Geometric measure theory, of course, but how much Riemannian geometry? How much partial differential equation theory for the theory of regularity as well? Thank you very much in advance!

Answer 1

The general prerequisites are almost the same as for currents, mainly a strong understanding of measure theory and a bit of geometrical intuition.

There is an aspect of multilinear algebra and some functional analysis involved as well, but a lot of that can be studied at the same time. The need for Riemannian geometry depends mostly on what you want to study. For varifolds on Riemannian manifolds, the need is kind of obvious, but for varifolds on $\mathbb{R}^n$, you can do completely without. The notion of mean curvature is involved in many problems involving varifolds, but not much more, as they simply lack the regularity needed for some of the more fancy ideas from differential geometry. Anything else, such as knowledge of PDEs and Calculus of variations is nice to have with regards to context, but the same is true in reverse and you have to start somewhere.

Fundamentally, varifolds in $\mathbb{R}^n$ are just Radon measures on $\mathbb{R}^n \times Gr(n,m)$, where $Gr(n,m)$ is the space of $m$-dimensional linear subspaces of $\mathbb{R}^n$. If you understand all the words in that sentence and know a bit about rectifiability, then you have the prerequisites to study varifolds.

P.S. I wouldn't say that they are either weaker or stronger than currents though. In the technical sense, there are currents that cannot be reasonably expressed as varifolds and varifolds that cannot be expressed as currents (even ignoring the issue of orientation). Similarly, in the practical sense, each has its advantages and its downsides and which one is better is entirely problem-dependent.