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In the previous work [35], the second and third authors established a Bochnertype formula on Alexandrov spaces. The purpose of this paper is to give someapplications of the Bochner type formula. Firstly, we extend the sharp lowerbound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6],from smooth Riemannian manifolds to Alexandrov spaces. As an application, weget an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp)Li-Yau's estimate for positve solutions of heat equations on Alexandrov spaces.

Sharp Spectral Gap and Li-Yau's Estimate on Alexandrov Spaces