Uniqueness and regularity of proper harmonic maps
Harmonic maps \(u:M^ m \to N^ n\) are critical points of the energy functional \(E(u)=\int_ Me(u)(x)dx\) where \(e(u)(x)\) is the energy density. The paper is devoted to th study of proper harmonic maps between hyperbolic spaces. One identifies hyperbolic space \(\mathbb{H}^ n\) via the Poincaré model with the unit ball \(D^ n \subset \mathbb{R}^ n\), the ideal boundary with \(S^{n-1}\), \(\overline D=\mathbb{H}^ n \cup S^{n-1}\). The first result is a uniqueness theorem. Theorem 1.1. Let \(u,v:\mathbb{H}^ m \to \mathbb{H}^ n\) be proper harmonic maps extending to \(C^ 1\)-maps \(u,v:\overline D^ m \to \overline D^ n\). Suppose that restricted to \(S^{m-1},u\) and \(v\) agree and that this boundary map has nowhere-vanishing energy density. Then \(u\) and \(v\) are identical. By constructing a solution of certain regularity for the Dirichlet problem and then applying the uniqueness theorem one obtains a regularity theorem. Theorem 4.3. Let \(u:\mathbb{H}^ m \to \mathbb{H}^ n\) be harmonic, so that \(u\) extends to a \(C^ 1\)-mapping \(u:\overline D^ m \to \overline D^ n\). Suppose, when restricted to \(S^{m-1}\), that the boundary map is in \(C^{k,\alpha}(S^{m-1},S^{n-1})\), for some \(1\leq k\leq m-1\), \(0<\alpha \leq 1\), and has nowhere-vanishing energy density. Then \(u \in C^{k,\gamma} (\overline D^ m,\overline D^ n)\) for all \(0<\gamma<\alpha\).
scientific article; zbMATH DE number 221042