A compactness result for energy-minimizing harmonic maps with rough domain metric (Q1688600)

In [Commun. Anal. Geom. 4, No. 1, 121--128 (1996; Zbl 0864.58014)] \textit{Y. Shi} generalized the \(\epsilon\)-regularity theorem of \textit{R. Schoen} and \textit{K. Uhlenbeck} [J. Differ. Geom. 17, 307--335 (1982; Zbl 0521.58021)] to energy-minimizing harmonic maps from a domain equipped with a Riemannian metric of class \(L^{\infty}\). In the present paper the authors obtain a compactness result for such energy minimizing maps. As an application they combine their result with Shi's theorem and obtain an improved bound on the Hausdorff dimension of the singular set under the assumption that the map has bounded energy at all scales. They also show that this last assumption can be removed when the target is simply connected.