Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps

Hirsch, Jonas; Stuvard, Salvatore; Valtorta, Daniele

doi:10.1090/tran/7595

Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps
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by Jonas Hirsch, Salvatore Stuvard and Daniele Valtorta PDF

Trans. Amer. Math. Soc. 371 (2019), 4303-4352 Request permission

Abstract:

In this paper we study the singular set of energy minimizing $Q$-valued maps from $\mathbb {R}^m$ into a smooth compact manifold $\mathcal {N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single-valued case, we prove that the target $\mathcal {N}$ being nonpositively curved but not simply connected does not imply continuity of the map.

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