Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLimits of $alpha$-harmonic maps
74
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 $1$.
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 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.
$alpha$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $alpha$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $alpha >1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $alpha$-harmonic maps for $alpha >1$ and then letting $alpha to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $alpha$-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By $varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $alpha$-Dirac-harmonic maps converges to a smooth nontrivial $alpha$-Dirac-harmonic map.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
