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

2015/06/19 by Da Rong Cheng, Cheng, Da Rong
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1506.06122

arxiv created 2015/06/19 · arxiv updated 2015/06/22

Abstract

In 1996, Shi generalized the epsilon-regularity theorem of Schoen and Uhlenbeck to energy-minimizing harmonic maps from a domain equipped with a bounded measurable Riemannian metric. 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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