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

In this paper we study the regularity of stationary and minimizing harmonic maps f: B₂(p) ⊆ M → N between Riemannian manifolds. If S^k(f) ≡ (x ε M: No tangent map at x is k + 1-symmetric} is the k^th-stratum of the singular set of f, then it is well known that dim Sk ≤ k, however little else about the structure of S^k(f) is understood in any gen-erality. Our first result is for a general stationary harmonic map, where we prove that S^k(f) is k-rectiflable. In fact, we prove for k-a.e. point x ε S^k(f) that there exists a unique k-plane V k ⊆ 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) ≤ n-3, is in fact n-3-rectiflable with uniformly finite n-3-measure. An effective version of this allows us to prove that jrfj has estimates in L³_weak, an estimate that is sharp as jrfj may not live in L³. More generally, we show that the regularity scale rf also has L³_weak estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications Sk Roughly, is the collection of points x ⊆ M for which no ball Br(x) is ε-close to being k + 1-symmetric. We show that Skε is k-rectiflable and satisfies the Minkowski estimate Vol. The proofs require a new L²-subspace approximation theorem for sta-tionary harmonic maps, as well as new W1;p-Reifenberg and rectifiable-Reifenberg type theorems. These results are generalizations of the clas-sical Reifenberg and give checkable criteria to determine when a set is k-rectiflable with uniform measure estimates. The new Reifenberg type theorems may be of some independent interest. The L²-subspace approxi-mation theorem we prove is then used to help break down the quantitative stratifications into pieces that satisfy these criteria.