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A Liouville-type theorem and Bochner formula for harmonic maps into metric spaces
Communications in Analysis and Geometry ( IF 0.7 ) Pub Date : 2021-01-08 , DOI: 10.4310/cag.2020.v28.n8.a4
Brian Freidin 1 , Yingying Zhang 2
Affiliation  

We study analytic properties of harmonic maps from Riemannian polyhedra into $\operatorname{CAT}(\kappa)$ spaces for $\kappa \in {\lbrace 0, 1 \rbrace}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into $\operatorname{CAT}(\kappa)$ spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an $L^p$ Liouville-type theorem for harmonic maps from admissible polyhedra into convex $\operatorname{CAT}(\kappa)$ spaces. Another consequence we derive from the target variation formula is the Eells–Sampson Bochner formula for $\operatorname{CAT}(1)$ targets.

中文翻译:

调和映射到度量空间的Liouville型定理和Bochner公式

我们研究了从黎曼多面体到$ \ kappa \ in中的\\ operatorname {CAT}(\ kappa)$空间中的谐波映射的解析性质,其中{\ lbrace 0,1 \ rbrace} $。在局部上,在域的每个高维面上,这相当于研究从平滑域到$ \ operatorname {CAT}(\ kappa)$空间的谐波映射。我们计算出一个捕获目标中的曲率边界的目标变化公式,并用它证明从可允许多面体到凸$ \ operatorname {CAT}(\ kappa)$空间的谐波映射的$ L ^ p $ Liouville型定理。我们从目标变化公式得出的另一个结果是$ \ operatorname {CAT}(1)$目标的Eells–Sampson Bochner公式。
更新日期:2021-01-10
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