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

Abstract

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.

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Authors

Aaron Naber

Northwestern University, Evanston, IL

Daniele Valtorta

University of Zürich, Zürich, Switzerland