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Register a new userA gap theorem for $alpha$-harmonic maps between two-spheres
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In this paper we consider approximations introduced by Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $alpha$-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the $alpha$-harmonic maps of degree $-1, 0$ or $1$.
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Critical points of approximations of the Dirichlet energy `{a} la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energies. Indeed, we prove that constant maps and the rotations of $S^2$ are the only critical points of $E_{alpha}$ for maps from $S^2$ to $S^2$ whose $alpha$-energy lies below some threshold. In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of $alpha$-harmonic maps.
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Holder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Holder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.
The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves their non-existence on S^2, obtains rigidity results for harmonic gradient vector fields on the two-sphere, classifies spherical quadratic gradient fields in all dimensions and determines the tension field, concluding with the discovery of a family of metrics making Hopf vector fields harmonic maps on S^{2n+1}.
We propose a new notion called emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $pto infty $. Infinity harmoncity appears in many familiar contexts. For example, metric projection onto the orbit of an isometric group action from a tubular neighborhood is infinity harmonic. Unfortunately, infinity-harmonicity is not preserved under composition. Those infinity harmonic maps that always preserve infinity harmonicity under pull back are called infinity harmonic morphisms. We show that infinity harmonic morphisms are precisely horizontally homothetic mas. Many example of infinity-harmonic maps are given, including some very important and well-known classes of maps between Riemannian manifolds.
In this paper, we will show the Yaus gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $kappa$, $kappageq0$, in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.
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