Harmonic maps into spaces with an upper curvature bound in the sense of Alexandrov (Q1433432)

The author studies harmonic maps into metric spaces satisfying the CAT(\(\kappa\)) comparison inequality (which are called here spaces of curvature bounded from above by \(\kappa\)). He develops the notion of order function for energy minimizing maps into such spaces and he generalizes several results which hold for energy minimizing maps into spaces with nonpositive curvature to this setting of spaces satisfying an upper curvature bound. The main results are the following: Theorem: let \((M,g)\) be a compact Riemannian manifold with or without boundary and let \((d_i)\) be a sequence of distance functions on a space \(X\) with curvature bounded from above by \(\kappa\). Assume that \(X\) is compact with respect to the topology induced by each \(d_i\), let \(h:M\to X\) be a continuous map and let \(f_i:M\to (X,d_i)\) be continuous energy minimizing maps in the homotopy class of \(h\) with \(f_i\) coinciding with \(h\) on \(\partial M\) if \(\partial M\not= \emptyset\). Let \(\delta_i\) be the pull-back distance of \(d_i\) under \(f_i\). If the energy of \(f_i\) is bounded from above by some constant that does not depend on \(i\) and if \(d_i\) converges uniformly to a distance function \(d_0\), then there exists a subsequence \((i'_i)\) of \((i)\) and an energy minimizing map \(f_0\) with respect to \(d_0\) so that \(f_{i'}\) converges pointwise to \(f\), \(\delta_{i'}(.,.)\) converges uniformly to \(d_0(f(.),f(.))\) and the energy of \(f_{i'}\) converges to that of \(f_0\). The result is an adaptation of a compactness theorem of \textit{N. J. Korevaar} and \textit{R. M. Schoen} [Commun. Anal. Geom. 5, No. 2, 333--387 (1997; Zbl 0908.58007)]. As an application, the author obtains the following: Theorem: Let \(S\) be a compact topological surface and let \(d\) be a distance function on \(S\) that makes it a metric space of curvature bounded from above by \(\kappa\), with the topology induced by the metric \(d\) being the topology of \(S\) as a surface. Let \(\Sigma_1\) be a compact Riemann surface of the same genus as \(S\) and \(\phi:\Sigma_1\to S\) a diffeomorphism. Then there exists an energy minimizing map \(f:\Sigma_1\to (S,d)\) in the homotopy class of \(\phi\), smooth metrics \(g_i\) on \(S\), and maps \(f_i\) which are energy minimizing diffeomorphisms with respect to \(g_i\), so that the pull back distance function of \(f_i\) converges uniformly to that of \(f\) and the energy density function of \(f_i\) converges to that of \(f\). Theorem: Let \(S\) be a compact topological surface and let \(d\) be a distance function on \(S\) which makes it a metric space of curvature bounded from above by \(\kappa\) and which induces a topology that is equivalent to the surface topology. Let \(\Sigma_1\) be a compact Riemann surface of the same genus as \(S\) and let \(f:\Sigma_1\to (S,d)\) be an energy minimizing map in its homotopy class. Then, for any \(P\in S\), the preimage \(f^{-1}(P)\) is either a point or a connected union of finite number of vertical arcs of the Hopf differential of \(f\). The last theorem generalizes a result of \textit{E. Kuwert} on harmonic maps into flat surfaces with cone type singularities of nonpositive curvature [Math. Z. 221, No. 3, 421--436 (1996; Zbl 0871.58028)].