In this paper we study the regularity of stationary and minimizing harmonic maps f: B2(p) ⊆ M → N between Riemannian manifolds. If Sk(f) ≡ (x ε M: No tangent map at x is k + 1-symmetric} is the kth-stratum of the singular set of f, then it is well known that dim Sk ≤ k, however little else about the structure of Sk(f) is understood in any gen-erality. Our first result is for a general stationary harmonic map, where we prove that Sk(f) is k-rectiflable. In fact, we prove for k-a.e. point x ε Sk(f) that there exists a unique k-plane V k ⊆ TxM 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 L3weak, an estimate that is sharp as jrfj may not live in L3. More generally, we show that the regularity scale rf also has L3weak 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.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty