A remark on gauge transformations and the moving frame method

Abstract

In this note we give a shorter proof of recent regularity results in [Riv07], [RS08]. We differ from the mentioned articles only in using the direct method of Hélein's moving frame to construct a suitable gauge transformation. Though this is neither new nor surprising, it enables us to describe a proof of regularity using besides the duality of Hardy-and BMO-space only elementary arguments of calculus of variations and algebraic identities. Moreover, we remark that in order to prove Hildebrandt's conjecture one can avoid the Nash-Moser imbedding theorem. There are no new results presented here, nor are there any techniques we could claim originality for.

Analyse des problèmes conformément invariants

“…Il s'agit de s'inspirer des travaux de Karen Uhlenbeck sur l'existence de jauges de Coulomb [52], qui consistent dans notre cas à faire une décomposition de Hodge non-linéaire de Ω. On pourra aussi consulter [45] pour une approche plus variationnelle du même résultat.…”

$$\varepsilon $$ ε -regularity for systems involving non-local, antisymmetric operators

“…In [Sch10a] the construction of P is done via minimization of E(P ) = P ∇P T + P ΩP 2 L 2 under the condition that P maps into SO(N ), a.e.. This is the argument that Hélein [Hél91] essentially used for his moving-frame technique, and it provides an alternative to Riviére's adaption of Uhlenbecks [Uhl82] gauge-theoretic construction of P in [Riv07].…”

“…In fact, the antisymmetry is shown to be closely related to the appearance of Hardy spaces, and also to Hélein's [Hél91] moving frame technique, cf. [Sch10a]. Motivated by this, Da Lio and Rivière [DLR11a] (for m = 1) showed that this regularizing effect of antisymmetry exists and appears also in the setting of m/2-harmonic maps, critical points of the energŷ Here, Ω ij ∈ L 2 (R m ) satisfies again (1.2), and |∇| α = (−∆) α 2 is the elliptic differential operator of differential order α with the symbol |ξ| α , for the precise definition we refer to Section A .…”

“…Both techniques can be extended to the fractional case, where Ω is still a pointwise multiplication [DLR11a,Sch11]. We adapt the arguments [Sch11,Sch10a] to this case of a non-local operator Ω[], by minimizing in Section 1.6 the energy…”

“…As well in the classical situation [Riv07], as also in the case of fractional harmonic maps, the argument relies on transforming the equation with an orthogonal matrix P (in a similar way as Hèlein's moving frame technique, cf. [Sch10a]). That is, one computes the respective equation P ∇u instead of ∇u, or P ∆ n 4 u instead of ∆ n 4 u and obtains a transformed Ω P , which for the right choice of P exhibits better properties than the original Ω: In the classical case, div(Ω P ) = 0, while in the fractional case, Ω P ∈ L 2,1 (where L 2,1 L 2 is the Lorentz space dual to the weak L 2 , denoted by L 2,∞ ).…”