Abstract
We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.
Highlights
Harmonic maps are critical points of the Dirichlet energy for mappings of Riemannian manifolds and are non-linear analogues of solutions to Laplace’s equation. They play an important role in geometry and one of the most fundamental aspects of the theory of harmonic maps, which is intimately connected to the geometry and topology of the codomain manifold N, is their regularity
The fact that harmonic maps are smooth on domains of dimension two is a consequence of a result of Rivière [34] asserting the continuity of critical points of functionals with conformally invariant Lagrangian on two dimensional domains
Motivated by Rivière’s regularity result, at least partially, Da Lio and Rivière [8] introduced the notion of fractional harmonic mappings of manifolds; on a domain of dimension one these maps are critical points of functionals which satisfy a type of conformal invariance
Summary
Harmonic maps are critical points of the Dirichlet energy for mappings of Riemannian manifolds and are non-linear analogues of solutions to Laplace’s equation. Motivated by Rivière’s regularity result, at least partially, Da Lio and Rivière [8] introduced the notion of fractional harmonic mappings of manifolds; on a domain of dimension one these maps are critical points of functionals which satisfy a type of conformal invariance. -harmonic u: N, which are critical points of I 0 with respect to inner and outer variations, using regularity theory for stationary free boundary harmonic maps v: Rm++1 → N up to the free boundary O He showed the maps he considered are smooth when m = 1 and smooth with the possible exception of a set of vanishing Hausdorff dimension m − 1 when m ≥ 2. In contrast to intrinsic fractional harmonic maps, the energies for which extrinsic fractional harmonic maps are critical points depend upon the choice of embedding of the target manifold N into some Euclidean space. Aside from the results we establish in this article, the only results regarding intrinsic fractional harmonic maps are those of Moser [30]
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