About the regularity of energy minimizing maps with values in a metric space of curvature bounded from above (Q2764826)

This paper considers energy minimizing maps from a Riemannian manifold \((M,g)\) into a complete locally compact metric space, and extends a (partial) regularity result of \textit{R. Schoen} and \textit{K. Uhlenbeck} [J. Differ. Geom. 17, 307-335 (1982; Zbl 0521.58021)]. NEWLINENEWLINENEWLINEThe unavoidable upper bound curvature condition on the target space is given by a bound \(K\) on the Alexandrov curvature, which is essentially a triangle comparison condition with a sphere of radius \(K\). NEWLINENEWLINENEWLINEOwing to the generality of the target space the energy of a map \(\varphi : (M^{m},g) \to (X,d)\) is defined, following \textit{M. Gromov} and \textit{R. Schoen} [Publ. Math., Inst. Hautes Étud. Sci. 76, 165-246 (1992; Zbl 0896.58024)], by: NEWLINE\[NEWLINEE(\varphi) = \limsup_{\varepsilon \to 0} \int_{M} \varepsilon^{-(2+m)} \int_{B_{\varepsilon}(x)} d(\varphi(x),\varphi(y)) dx dy NEWLINE\]NEWLINE The main result is that from a Riemannian manifold, of dimension at least 3, to a complete locally compact metric space with curvature bounded from above (by \(K>0\)), an energy minimising map is continuous except on a set of Hausdorff dimension less than or equal to \(m-2\). NEWLINENEWLINENEWLINEThe main tools are convergence theorems for blow-up sequences of the target space, for the Gromov-Hausdorff distance, and, in \(L^2\), for the blow-up of maps.