Flat norm of currents and minimal surfaces

Question

Let $A$ be a $k \leq n$ integral current with compact support over $\mathbb{R}^n$ (for conciseness). Its flat norm $F(A)$ can be defined via

$ F(A) = \inf \{ M(T) + M(S) \, | A = T + \partial S \}$

Yet after many discussions with competent people, the intuition behind the flat norm I've come to understand is that it somehow computes minimal surfaces. That is, if $A$ is a cycle, we in fact have $F(A) = \inf \{ M(S) \, | \, \partial S = A \}$

After some research on the internet, I found this theorem in a lecture from Alexis Michelat which seems to justify this idea when flat norm of the current is "small enough", but nothing more.

Is this intuition justified ? Are there counter-examples, and if so, are they pathological ? Would my statement be true for some "nice" subclass of integral currents ?

Answer 1

I think you've got a few things slightly mixed up, which made your question a bit unclear.

I don't like the heuristic that the flat norm 'computes minimal surfaces'. For one, it's not true that the flat norm of a boundary is always realized by a minimal surface. That is, in general you only have the inequality \begin{equation} F(A) \leq \inf \, \{ \mathbf{M}(T) \mid \partial T = A \}. \end{equation}

For example, take a circle $C_R \subset \mathbf{R}^2$ of radius $R > 0$ in the Euclidean plane. Let $A = C_R$ be this circle. This spans only one minimal surface, namely the disk $D_R$. However, if you compare their masses, you find that $\mathbf{M}(C_R) = \mathcal{H}^1(C_R) = 2 \pi R$, while $\mathbf{M}(D_R) = \mathcal{H}^2(D_R) = \pi R^2$.

You will find the same behavior if you rescale an arbitrary boundary $A = \partial Q$. This is because there is a second inequality, which 'competes' with the one above: \begin{equation} F(\partial Q) \leq \mathbf{M}(\partial Q). \end{equation} If you rescale $\partial Q$ by $\lambda > 1$, then the right-hand side changes like \begin{equation} \mathbf{M}(\lambda \cdot \partial Q) = \lambda^k \mathbf{M}(\partial Q), \end{equation} whereas the bound from the inequality grows quicker: \begin{equation} \inf \, \{ \mathbf{M}(T) \mid \partial T = \lambda \cdot \partial Q \} = \lambda^{k+1} \inf \, \{ \mathbf{M}(T) \mid \partial T = \partial Q \}. \end{equation} Therefore, if you take $\lambda$ large enough, you will get the strict inequality \begin{equation} F(\lambda \cdot \partial Q) < \inf \, \{ \mathbf{M}(T) \mid \partial T = \lambda \cdot \partial Q \}. \end{equation} (In general it's a good idea with these things to check how they behave under homothetic rescalings.)

To get a better feel for the flat norm, here are some questions that should be helpful to think about.

What is the flat norm of two disks in $\mathbf{R}^3$, lying above one another? How does it depend on the radius, and the distance between them? Does it depend on their orientations? What if they are given multiplicities?

In general, I would suggest spending some time thinking about the flat distance. For example, say the disks above lie a distance $1/j$ apart. What is their limit (with respect to the flat distance) as $j \to \infty$? Does it depend on their orientations, or possible multiplicities?

Another potentially useful example you get by considering the graph of a function $u: D_1 \to \mathbf{R}$, denoted $G$. Assuming $G$ is close to $D_1 \times \{ 0 \}$ in the flat distance, what information is gained on $u$? Does it give you a bound almost everywhere, in $L^p$, etc.? Does it give you a bound for the area of $G$? If you have a sequence $(u_j)$, and $G_j \to D_1 \times \{ 0 \}$ as $j \to \infty$, does this mean $u_j \to 0$? And in what topology?

As far as references are concerned, I am assuming you are aware of Leon Simon's lecture notes on GMT. You could also take a look at Frank Morgan's book; also I believe Otis Chodosh has some lecture notes from a course taught by Brian White that you might enjoy.

Answer 2

Leo Moos gave a very good answer, but here is another way to think about this:

Essentially equality does not hold when it is cheaper to have a the boundary of a hole than filling that hole. So if $S$ would be the minimal surface, with $\partial S = A$ then the question is, can we "cut out" a $S'$ (in the sense that $M(S)= M(S-S')+M(S')$ ), such that $M(S') > M(\partial S')$? Because then $A= \partial (S-S') + (-\partial S')$ obviously is a better candidate for the flat norm.

Now assuming that $S'$ is integer rectifiable¹, then a quick look at the isoperimetric inequality shows that this can only work if $M(S')$ and thus $M(S)$ is large enough. After all, $S'$ should also be a minimal surface to its boundary and then $M(S') \leq \gamma M(\partial S')^{(k+1)/k}$.

As another side-note, it is perfectly valid and sometimes useful to define a "homogenized flat norm", ommitting the $T$-term. As long as one restricts $A$ to a class where it is not infinite (e.g. $\partial A = 0$ in $\mathbb{R}^n$, or generally anything with the right homology), then it has more or less the same properties as the flat norm.²

¹Already $S$ does not have to be rectifiable, even if $A$ is. This is a completely different problem, that can prevent $F$ from resulting in a minimal surface. It is also one of the reasons, why in most texts one defines both $F$ as the infimum over arbitrary currents and $\mathcal{F}$ as the infimum over integer-rectifiable currents. The latter looses the nice duality structure, but avoids this issue.

²If you are interested in function spaces, you can also compare the dual $(W^{1,\infty}_0)^*$ and how it changes depending on if you choose $\|u\|_{W^{1,\infty}_0}= \sup | \nabla u|$ or $\|u\|_{W^{1,\infty}_0} = \sup |u| + \sup |\nabla u|$. This is actually equivalent to the $0$-current case and just looking at Dirac-deltas already reveals a lot of the differences in behaviour.