just saw this on the arxiv today, off a preprint of n. katzourakis. it's about infinite-harmonic maps from the plane into space, but it was this result that piqued my curiosity [1]:

interesting:

[1]

**Theorem**(κatzοurakis). Suppose $\Omega \subseteq \mathbb{R}^n$ is open and contractible and $u : \Omega \to \mathbb{R}^N$ is in $C^2(\Omega)^N$. Then the following are equivalent:- $u$ is a Rank-One map, that is ${\rm rank}(Du) \leq 1$ on $\Omega$, or equivalently there exist $\xi: \Omega \to \mathbb{R}^N$ and $w : \Omega \to \mathbb{R}^n$ such that $Du = \xi \otimes w$;
- there exists $f \in C^2(\Omega)$, a partition $\{B_i\}_{i=1}^\infty$ of $\Omega$ of Borel sets, where each $B_i$ equals a connected open set with a boundary portion and Lipschitz curves $\{\nu^i\}_{i=1}^\infty$ in $W^{1,\infty}_{\rm loc}(\mathbb{R})^N$ such that on each $B_i$, $u$ equals the composition of the curve $\nu^i$ with the scalar function $f$:

$$u \;=\; \nu^i \circ f, \hspace{.5in} \text{ on } B_i \subseteq \Omega$$Moreover, $|\dot{\nu}^i| \equiv 1$ on $f(B_i)$, $\dot{\nu}^i \equiv 0$ on $\mathbb{R} \setminus f(B_i)$, and there exist $({\nu}^i)''$ on $f(B_i)$, interpreted as $1$-sides on $\partial f(B_i)$, if any. Also,

$$ Du \;=\; (\dot{\nu}^i \circ f) \otimes Df, \hspace{.5in} \text{ on } B_i \subseteq \Omega$$and the image $u(\Omega)$ is a $1$-rectifiable subset of $\mathbb{R}^N$.

interesting:

roughly speaking, if the derivative tells you that a smooth mapping has 1-dimensional behavior, then you can actually cut it up into athere's also a version purely for maps with components in $W^{1,\infty}(\Omega)^N$, in the same paper, but with an $L^\infty$-approximation condition via smooth maps.singlefunction that mimicks the mapping's behavior through curves.

(i haven't read the proof, but my guess is that the hard work is somehow done through the partitioning. i wonder if there is a Rank-$M$ version of this result, for $M \in \mathbb{N}$ ..)

[1]

*the blue text was added today (1 May 2012). i could swear that it was there before, but it seemed to have disappeared after reloading the blog page. also, i decided to indent some of the last few paragraphs to highlight my guess on the subject.*
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