Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps


In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is the $k^{\rm th}$-stratum of the singular set of $f$, then it is well known that $\dim S^k\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable. In fact, we prove for $k$-a.e. point $x\in S^k(f)$ that there exists a unique $k$-plane $V^k\subseteq T_xM$ such that every tangent map at $x$ is $k$-symmetric with respect to $V$.
In the case of minimizing harmonic maps we go further and prove that the singular set $S(f)$, which is well known to satisfy $\dim S(f)\leq n-3$, is in fact $n-3$-rectifiable with uniformly finite $n-3$-measure. An effective version of this allows us to prove that $|\nabla f|$ has estimates in $L^3_{\rm weak}$, an estimate that is sharp as $|\nabla f|$ may not live in $L^3$. More generally, we show that the regularity scale $r_f$ also has $L^3_{\rm weak}$ estimates.
The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications $S^k_{\epsilon,r}(f)$ and $S^k_{\epsilon}(f)\equiv S^k_{\epsilon,0}(f)$. Roughly, $S^k_{\epsilon}\subseteq M$ is the collection of points $x\in M$ for which no ball $B_r(x)$ is $\epsilon$-close to being $k+1$-symmetric. We show that $S^k_\epsilon$ is $k$-rectifiable and satisfies the Minkowski estimate $\mathrm{Vol}(B_r\,S_\epsilon^k)\leq C r^{n-k}$.
The proofs require a new $L^2$-subspace approximation theorem for stationary harmonic maps, as well as new $W^{1,p}$-Reifenberg and rectifiable-Reifenberg type theorems. These results are generalizations of the classical Reifenberg and give checkable criteria to determine when a set is $k$-rectifiable with uniform measure estimates. The new Reifenberg type theorems may be of some independent interest. The $L^2$-subspace approximation theorem we prove is then used to help break down the quantitative stratifications into pieces that satisfy these criteria.


Aaron Naber

Northwestern University, Evanston, IL

Daniele Valtorta

University of Zürich, Zürich, Switzerland