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

TL;DR

This paper proves a compactness theorem for energy-minimizing harmonic maps with rough domain metrics, improving understanding of their singular sets and extending prior regularity results.

Contribution

It establishes a new compactness result for harmonic maps from domains with measurable metrics, enhancing regularity theory and singular set estimates.

Findings

Proves a compactness theorem for energy-minimizing harmonic maps with rough metrics.

Provides improved bounds on the Hausdorff dimension of singular sets.

Extends regularity results to maps with bounded energy at all scales, especially for simply-connected targets.

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.