Bochner  inequality is one of the most important tools in geometric analysis.  In this talk, we will first introduce some version of Bochner inequality for functions,  or harmonic maps from Alexandrov spaces with curvature bounded from below (or RCD-spaces) into CAT(1)-spaces.  As an application, the Cheng-Yau’s gradient estimate for these harmonic maps were given. This is a joint work with Xiao Zhong and Xi-Ping Zhu.

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