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

In 1996, Shi [6] generalized the $\epsilon$-regularity theorem of Schoen and Uhlenbeck [5] to energy-minimizing harmonic maps from a domain equipped with a Riemannian metric of class $L^{\infty}$. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi’s theorem to give an improved bound on the Hausdorff dimension of the singular set, assuming that the map has bounded energy at all scales. This last assumption can be removed when the target manifold is simply-connected.

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Received 19 June 2015

Published 30 November 2017