On the Size of the Singular Set of Minimizing Harmonic Maps

We consider minimizing harmonic maps u u from Ω ⊂ R n Ω ⊂ R^n into a closed Riemannian manifold N N and prove: 1. an extension to n ≥ 4 n ≥ 4 of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold N N is finite, we have \[ H n − 3 ( sing ⁡ u ) ≤ C ∫ ∂ Ω | ∇ T u | n − 1 d H n − 1 ; Hⁿ⁻³(sing u) ≤ C ∫ ∂ Ω |∇ _T u|ⁿ⁻¹ \,dHⁿ⁻¹; \] 2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is N = S 2 N=S^2 we obtain that the singular set of u u is stable under small W 1 , n − 1 W1,n-1 -perturbations of the boundary data. In dimension n = 3 n=3 both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space W s , p Ws,p with s ∈ ( 1 2 , 1 ] s ∈ (1/2,1] and p ∈ [ 2 , ∞ ) p ∈ [2,∞ ) satisfying s p ≥ 2 sp ≥ 2 . We also discuss sharpness.