Reference request: Intuitive introduction to currents and varifolds

Question

As I have recently been interested in geometric measure theory related problems, I am learning some of the basics of the field. I am looking for a textbook that introduces currents and varifolds in an intuitive and pedagogical manner for beginners to the subject. I know of Leon Simon’s Lectures on Geometric Measure Theory, which is really good for the precise details but I am finding some of the intuition hard to grasp.

Answer 1

It's difficult to find an elementary level introduction to currents and varifolds, but possibly the most beautiful (very) elementary introduction that you need to grasp the feel of the subject are the beautiful lecture notes [1a] [1b] of Almgren. Be warned that there are two major "shortcomings" in these presentation:

• the definition of varifold is not the modern one put forward by William K. Allard (even if it is equivalent to it), and
• it deal "only" with subsets of $\mathbf R^3$ (no manifolds or more general ambient spaces).

Nevertheless it is still, in my opinion, an extremely useful introduction: just to see an example of this, let's compare respectively the definitions of currents and varifolds given there (I tried to update a bit the notation, hopefully to clarify the meaning). Let $\Omega^k\equiv\Omega^k(\mathbf R^3)$ be the space of $k$-differential forms on $\mathbf R^3$ and $\wedge$ be their standard exterior product.

Definition 1. ([1a] chapter 3, §3.1 p. 36-37, [1b] chapter 3, §3.1 p. 37-38) A $k$-dimensional current in $\mathbf R^3$ is a linear functional $A:\Omega^k\to\mathbf R$ such that

1. $A$ is linear, i.e. $$ \begin{align} A(r\varphi)& =r A(\varphi) \\ A(\varphi+\psi) &= A(\varphi) + A(\psi) \end{align}\quad\forall r\in\mathbf R, \forall \varphi ,\psi\in\Omega^k. $$
2. $A$ is continuous respect to the standard topology of $\Omega^k$.

Definition 2. ([1a] chapter 3, §3.1 p. 37-38, [1b] chapter 3, §3.1 p. 38-39) A varifold in $\mathbf R^3$ is a positive functional $\|B\|:\Omega^k\to\mathbf R_{\ge 0}$ such that

1. $\|B\|$ is positively homogeneous of degree $1$, i.e. $$ \|B\|(r\varphi) = |r| \|B\|(\varphi) \quad\forall r\in\mathbf R, \forall \varphi \in\Omega^k. $$
2. $\|B\|$ is sub-additive, i.e. $$ \|B\|(\varphi+\psi) \le \|B\|(\varphi) + \|B\|(\psi)\quad\forall \varphi ,\psi\in\Omega^k. $$
3. For each continuous function $f, g:\mathbf R^3\to\mathbf R_{\ge 0}$ we have $$ \|B\|((f+g)\wedge\varphi) = \|B\|(f\wedge\varphi) + \|B\|(g\wedge\varphi) \quad \forall \varphi \in\Omega^k. $$

Note that the first two conditions in definition 2 are a generalisation of the condition of linearity required in definition 1, while the third one is a generalisation of the continuity condition, which is however implied by it and left as an exercise ([1a] chapter 3, §3.3 p. 43, [1b] chapter 3, §3.3 p. 44).

Regarding the intuition behind the concept, I can say that Almgren is careful in explaining ([1a] chapter 3, §3.1 p. 36, [1b] chapter 3, §3.1 p. 37 and, respectively, [1a] chapter 3, §3.1 p. 37, [1b] chapter 3, §3.1 p. 38) that the integration of a differential form $\varphi$ on a rectifiable set $A\Subset\mathbf R^3$, i.e. $$ A(\varphi) = \int_A\varphi \quad \forall \varphi \in\Omega^k $$ is the evaluation of a current according to definition 1 above, while performing the same operation on a rectifiable $B\Subset\mathbf R^3$ ([1a] chapter 2, §2.8, [1b] chapter 2, §2.8) but disregarding the orientation on $B$, in his notation $$ \|B\|(\varphi)=\int_B\|\varphi \|\quad \forall \varphi \in\Omega^k, $$ is de facto the evaluation a (rectifiable) varifold according to definition 2 above. Thus varifolds extend the notion of integral of a differential form on a given set by dropping the need of considering the orientation on that set, which thus in this way can be undefined or simply not existing.

This is the most elementary treatment of the subject I'm aware of: if you need something more advanced then I think Simon's Lectures you are already looking at are the best you can find.

Reference

[1a] Frederick Justin Almgren, jun. Plateau’s problem. An invitation to varifold geometry, New York-Amsterdam: W.A. Benjamin, Inc., pp. XII+74 p. (1966), MR190856, Zbl 0165.13201.

[1b] Frederick Justin Almgren, jun. Plateau’s problem. An invitation to varifold geometry, revised ed., Student Mathematical Library, 13, Providence, RI: American Mathematical Society (AMS), pp. xvi+78 (2001), ISBN:0-8218-2747-2, MR1853442, Zbl 0995.49001.