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Register a new userApproximate solutions to the Dirichlet problem for harmonic maps between hyperbolic spaces
الحلول التقريبية لمشكلة ديريشل لخرائط الهارمونية بين المساحات الهيبروبولية
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نتيجتنا الرئيسية في هذا البحث هي التالية: بفرض $H^m, H^n$ هي مساحات هيبروليكية للأبعاد $m$ و $n$ المتطابقة، وبفرض دالة Holder $f=(s^1,...,f^{n-1}):partial H^mto partial H^n$ بين الحدود الهندسية لـ $H^m$ و $H^n$. ثم لكل $epsilon >0$ يوجد خريطة هارمونية $u:H^mto H^n$ التي تكون متواصلة حتى الحدود (بالأحوال الإيوكليدية) و $u|_{partial H^m}=(f^1,...,f^{n-1},epsilon)$.
Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):partial H^mto partial H^n$ between geometric boundaries of $H^m$ and $H^n$. Then for each $epsilon >0$ there exists a harmonic map $u:H^mto H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{partial H^m}=(f^1,...,f^{n-1},epsilon)$.
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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.
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.
Let ${u_n}$ be a sequence of maps from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold $N$ with free boundary on a smooth submanifold $Ksubset N$ satisfying [ sup_n left(| abla u_n|_{L^2(M)}+|tau(u_n)|_{L^2(M)}right)leq Lambda, ] where $tau(u_n)$ is the tension field of the map $u_n$. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time.
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}.
We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.
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