Stationary harmonic maps into complete Riemannian manifold (Q1974196)

Let \((M,g)\) be a smooth compact Riemannian manifold of dimension \(n\), and \((N,h)\) be a smooth complete noncompact Riemannian manifold of dimension \(m\) which is isometrically embedded into Euclidean space \(\mathbb{R}^k\). Let \(H^1(M;N)= \{u\in H^1(M;\mathbb{R}^k):u(x)\in N\) for a.e. \(x\in M\}\). In this paper the author studies the singular set of a stationary map \(u\in H^1 (M;N)\) in the sense of Schoen [see \textit{R. Schoen}, Publ., Math. Sci. Rec. Inst. 2, 321-358 (1984; Zbl 0551.58011)]. He proves the following theorem: Suppose \(M\) is flat and is bounded by a bounded domain \(\Omega\) contained in \(\mathbb{R}^n\). Let \(u\in H^1(M;N)\) be a stationary map. Then there is a closed set \(Q\subset \Omega\) such that \(u\in C^\infty (M\setminus Q;\mathbb{R}^k)\), where \(Q=Q_1\cup Q_3\), and \(Q_1=\{x\in \Omega:\lim_{r\to 0}\sup r^{2-n} \int_{B_r(x)} |\nabla u|^2 >0\}\), and \(Q_3=\{x\in M:\lim_{r\to 0}N(x,r,u)=0\}\), in which \(B_r(x)= \{y\in R^n:|y-x|<r\}\subset \Omega\), \(N(x,r,u)=\sup \{|u|_{B (y,\rho)}: B(y,\rho) \subset B_r(x)\}\). When \(u\) is an energy-minimizing map into a noncompact complete Riemannian manifold \(N\), \textit{M. Li} [Calc. Var. Partial Differ. Equ. 3, 513-529 (1995; Zbl 0920.58023)] has studied its regularity. For the study of singular set of a stationary map also see \textit{L. C. Evans} [Arch. Ration. Mech. Anal. 116, 101-113 (1991; Zbl 0754.58007)] and \textit{F. Bethuel} [Manuscr. Math. 78, 417-443 (1993; Zbl 0792.53039)].