Minimizing 1/2-harmonic maps into spheres

Abstract

In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of [30,33] in the case where the target manifold is the (m − 1)-dimensional sphere. For m 3, we show that minimizing 1/2-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For m = 2, we prove that, up to an orthogonal transformation, x/|x| is the unique non trivial 0-homogeneous minimizing 1/2-harmonic map from the plane into the circle S 1 . As a corollary, each point singularity of a minimizing 1/2-harmonic maps from a 2d domain into S 1 has a topological charge equal to ±1.

Partial Regularity for Fractional Harmonic Maps into Spheres

“…In the case s = 1/2, sequences of (stationary or not) 1/2-harmonic maps are not compact in general, see e.g. [6,31,35,30]. The prototypical example is the following sequence of smooth 1/2-harmonic maps from R n into S 1 ⊆ C given by…”

“…Since each u k is minimizing, we infer from [30,Theorem 3.6] that each u e k is a minimizing harmonic map with (partially) free boundary in G , i.e., [30,Definition 3.1]). Applying [30, Theorem 3.5] (which is based on [13,14]), we conclude that u e k → u e strongly in H 1 loc (G ∪ ∂ 0 G ; R d ), and that u e is a minimizing harmonic map with (partially) free boundary in G .…”

“…for critical points of higher order or/and p-power type energies (still in the corresponding critical dimension), see [7,11,12,29,44,45,46]. The regularity theory for 1/2-harmonic maps into a manifold in higher dimensions has been addressed in [36] and [32] (see also [30]). In higher dimensions, the theory provides partial regularity (i.e.…”

“…15. This example is built on the minimality of the map x/|x| from the plane R 2 into S 1 proved in [30,Theorem 1.4]. The minimality of x/|x| for s = 1/2 is open, but one can check that it is at least a stationary s-harmonic map into S 1 in the unit disc D 1 ⊆ R 2 , showing that Theorem 1.2 is sharp also for s ∈ [1/2, 1).…”

“…For s = 1/2 and d 3 (i.e., for S 2 or higher dimensional target spheres), the size of the singular set of a minimizing 1/2-harmonic map can be reduced. It has been proved in [30,Theorem 1.3] that in this case, sing(u) = ∅ for n = 2, it is locally finite for n = 3, and dim H sing(u) n − 3 for n 4. It would be interesting to know if this improvement persists for s = 1/2.…”