Locally minimizing harmonic maps from noncompact manifolds (Q1344913)

Let \((M,g)\) denote a complete non-compact Riemannian manifold and consider a compact manifold \(N\) isometrically embedded in some Euclidean space \(\mathbb{R}^ k\). Suppose further that the spectrum of the Laplacian of \((M,g)\) has positive infimum. Then given \(\Phi \in C^ 2(M,N)\) satisfying \(| \nabla \Phi(x)| \to 0\) as \(| x| \to \infty\) and the norm of the tension field is in \(L^ 2(M)\), the author constructs a harmonic map \(f : M \to N\) such that \[ \int_ M \text{dist} (f,\Phi)^ 2 < \infty \quad \text{and}\quad f - \Phi \in H^{1,2} (M,\mathbb{R}^ k). \] Moreover, the regularity results of Schoen and Uhlenbeck apply to \(f\).