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Register a new userOn the size of the singular set of minimizing harmonic maps
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We consider minimizing harmonic maps $u$ from $Omega subset mathbb{R}^n$ into a closed Riemannian manifold $mathcal{N}$ and prove: (1) an extension to $n geq 4$ of Almgren and Liebs linear law. That is, if the fundamental group of the target manifold $mathcal{N}$ is finite, we have [ mathcal{H}^{n-3}(textrm{sing } u) le C int_{partial Omega} | abla_T u|^{n-1} ,d mathcal{H}^{n-1}; ] (2) an extension of Hardt and Lins stability theorem. Namely, assuming that the target manifold is $mathcal{N}=mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s in (frac{1}{2},1]$ and $p in [2,infty)$ satisfying $sp geq 2$. We also discuss sharpness.
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We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n geq 4$. For minimizing harmonic maps $uin W^{1,2}(Omega,mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Liebs linear law, namely [mathcal{H}^{n-3}(textrm{sing} u) le C int_{partial Omega} | abla_T u|^{n-1} ,dmathcal{H}^{n-1};] (2) An extension of Hardt and Lins stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.
This article addresses the regularity issue for minimizing fractional harmonic maps of order $sin(0,1/2)$ from an interval into a smooth manifold. Holder continuity away from a locally finite set is established for a general target. If the target is the standard sphere, then Holder continuity holds everywhere.
We discuss a method to construct Dirac-harmonic maps developed by J.~Jost, X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. textbf{59} (2009), no. 11, 1512--1527.The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension $1$ with target spaces of constant sectional curvature.Before the present article, it remained unclear when the conditions of the theorems in J.~Jost, X.~Mo, M.~Zhu, emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. textbf{59} (2009), no. 11, 1512--1527, were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions.In several cases we show the existence of solutions.
In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $mgeq 3$, 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 $mathbb{S}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d domain into $mathbb{S}^1$ has a topological charge equal to $pm1$.
Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.
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