ترغب بنشر مسار تعليمي؟ اضغط هنا

Lipschitz continuity of harmonic maps between Alexandrov spaces

295   0   0.0 ( 0 )
 نشر من قبل Hui-Chun Zhang
 تاريخ النشر 2013
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

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 theor ems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.
In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in cite{lin97}. The proof extends the argument of Huang-Wang cite {hua-w10}.
In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li-Yaus estimate for positve solutions of heat equations on Alexandrov spaces.
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 i dentity in [7], we obtain an optimal gap theorem for the $alpha$-harmonic maps of degree $-1, 0$ or $1$.
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 energi es. 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.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا