Free boundary minimal surfaces: a nonlocal approach

Existence, Uniqueness and Regularity of the Fractional Harmonic Gradient Flow in General Target Manifolds

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In this paper, we continue to study the fractional harmonic gradient flow on S 1 taking values in a general closed manifold N ⊂ R n , addressing global existence and uniqueness of solutions of energy class with sufficiently small energy, adding to the existing body of knowledge pertaining to the half-harmonic gradient flow and expanding upon our previous work in [34]. We extend the techniques by Struwe in [30] and Rivière in [22] to the non-local framework analogous to [34] to derive uniqueness, employ commutator estimates as in [8] for regularity and follow [30] for a general existence result.

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Half-Harmonic Gradient Flow: Aspects of a Non-Local Geometric PDE

“…A crucial observation is that ϕ is supported on a subdomain of B 2 N /Rn for R n sufficiently small, so the estimates have good bounds everywhere, if n goes to ∞. So we are done, since v solves the half-harmonic map equation and thus is actually smooth, see [6]. In particular, v may be regarded as a 1/2-harmonic map after composition with the stereographic projection.…”

“…Fractional Laplacians (−∆) s/2 may be defined by Fourier multipliers or using principal value integrals, we refer to [17] for some exposition or the next subsection where both kinds of definitions are introduced. These maps are related to free-boundary minimal discs ( [7], [6]) and singular limits of Ginzburg-Landau approximations ( [16]), so there are again interesting connections to geometry.…”

“…Then, localising and applying Wente-type estimates, we arrive again at a Morrey decrease and, by Adams embedding, at higher integrability and thus Hölder regularity. Following [6], one may bootstrap this information to arrive at smoothness of solutions.…”

Minimizing 1/2-harmonic maps into spheres

“…(Note that this proposition is stated for N = S 1 , but the proof actually applies to any target manifold N .) It has been (independently) proved in [3,9,10,12], and [30, Lemma 4.27 & Remark 4.29] that g being 1/2-harmonic implies that w g is (weakly) conformal or anticonformal, i.e., it satisfies…”