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

• 成果类型: Article
• 署名作者: Naber, Aaron; Valtora, Daniele
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.185.1.3
• 发表日期: 2017
• 页码: 131-227
• 关键词: varying topological type; q-valued functions; quantitative stratification; SINGULAR SET; n-rectifiability; EXISTENCE; SURFACES; currents; spheres; terms

摘要

In this paper we study the regularity of stationary and minimizing harmonic maps f: B-2(p) subset of M -> N between Riemannian manifolds. If Sk(f)={x is an element of M: no tangent map at x is k+1-symmetric} is kth-stratum of the singular set of f, then it is well known that dimSk=k, however little else about the structure of Sk(f) is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that Sk(f) is k-rectifiable. In the case of minimizing harmonic maps we go further, and prove that the singular set S(f), which is well known to satisfy dimS(f)<= n-3, is in fact n-3-rectifiable with uniformly {\it finite} n-3-measure. An effective version of this allows us to prove that |del f| has estimates in L-weak(3), an estimate which is sharp as |del f| may not live in L3. The above results are in fact just applications of a new class of estimates we prove on the {it quantitative} stratifications S-epsilon,r(k)(f) and S-epsilon(k)(f)=S-epsilon,0(k),(f). Roughly, S-epsilon,k(k) subset of M is the collection of points x is an element of Sk? for which no ball B-r(x) is epsilon-close to being k+1-symmetric. We show that S-epsilon(k) is k-rectifiable and satisfies the Minkowski estimate Vol(BrS epsilon k)<= Crn-k. The proofs require a new L-2-subspace approximation theorem for stationary harmonic maps, as well as new W-1,W-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.