On the Dirichlet problem for harmonic maps with prescribed singularities
In this paper, the Dirichlet problem for harmonic maps with prescribed singularities from a smooth bounded domain \(\Omega \subset \mathbb{R}^n\), \(n \geq 2\), into \((M,g)\), a classical Riemannian globally symmetric space of rank one and of noncompact type is studied. Thus \((M,g)\) is either the real-, complex-, or quaternion-hyperbolic space, i.e., \((M,g) = \mathbb{H}^\ell_\mathbb{K}\), where \(\ell \geq 2\), and \(\mathbb{K}\) is either \(\mathbb{R}\), \(\mathbb{C}\), or the quaternions \(\mathbb{H}\). All results obtained apply to bounded domains in Riemannian manifolds. The main result may be stated as follows. Let \(\Sigma_i\), \(i = 1, \dots, N\), be disjoint closed smooth submanifolds of \(\Omega\) of codimension at least 2, and set \(\Sigma = \bigcup^N_{i = 1} \Sigma_i\). For each \(1 \leq i \leq N\), let \(\gamma_i : \mathbb{R} \to (M,g)\) be a unit speed geodesic, and let \(\varphi_i : \Omega \setminus \Sigma_i \to (M,g)\) be a harmonic map singular on \(\Sigma_i\) whose image is contained in \(\gamma_i (\mathbb{R})\) and such that \(\varphi (x) \to \gamma_i (+\infty) \in \partial M\) as \(x \to \Sigma_i\). Such a map is called \(\Sigma_i\)- singular map into \(\gamma_i\), provided it satisfies an additional technical condition. Since \(\gamma_i\) is (trivially) flat and totally geodesic, such a map is easily constructed from a harmonic function \(u_i\) on \(\Omega \setminus \Sigma_i\) which tends to infinity on \(\Sigma_i\). Let \(\psi : \partial \Omega \to M\) be a smooth boundary map. The result obtained is that there exists a unique harmonic map \(\varphi : \Omega \setminus \Sigma \to (M,g)\) which has boundary values \(\psi\), and is asymptotic to \(\varphi_i\) near \(\Sigma_i\).
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