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Asymptotic boundary value problem of harmonic maps via harmonic boundary. (English) Zbl 1300.58006

In this paper, the author proves the existence of harmonic map between complete Riemannian manifolds taking the given boundary data on the harmonic boundary of the domain manifold. Let \(M\) be a complete Riemannian manifold and let \(\widehat M\) be its Royden compactification, i.e., \(\widehat M\) is a locally compact Hausdorff space which contains \(M\) as an open dense subset. Then the harmonic boundary \(\Delta_M\) of \(M\) is defined by \[ \Delta_M = \{\mathbf{x}\in \partial \widehat M\,:\, f(\text\textbf{x}) = 0 \,\, \text{for all \(f \in {\mathcal {BD}_0}(M)\)}\}, \] where \(\partial \widehat M = \widehat M\setminus M\) is the Royden boundary and \({\mathcal {BD}_0}(M)\) is the closure of the set of all compactly supported smooth functions in the space of all bounded continuous functions on \(M\) whose distributional gradient \(\nabla f\) belongs to \(L^2(M)\). Let \(B_r(p)\) be a geodesic ball in another complete Riemannian manifold \(N\) which lies within the normal range of \(p\) in \(N\). The author shows that if \(f \in C(\Delta_M, B_r(p))\), there exists a harmonic map \(u \in C(M, B_r(p))\) such that for each \(\mathbf{x}\in \Delta_M\), \[ \lim_{x\in M \to \mathbf{x}} u(x) = f(\mathbf{x}). \]

MSC:

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
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