On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher

2019/02/08 by Mazowiecka, Katarzyna, Miśkiewicz, Michał, Schikorra, Armin · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.03161

Abstract

We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions n ≥ 4. For minimizing harmonic maps u∈ W^1,2(Ω,\mathbbS²) from n-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Lieb's linear law, namely H^n-3(\textrmsing u) ≤ C ∫_{∂ Ω} |∇_T u|^n-1 dH^n-1; (2) An extension of Hardt and Lin's stability theorem, namely that the size of singular set is stable under small perturbations in W^1,n-1 norm of the boundary.

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