In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau’s gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

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Inspired by the work of Z. Lu and G. Tian (Duke Math. J. 125:351–387,2004) in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan–Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan–Hartogs domain can be viewed as an “iterated” disk bundle over its base and the ideas given in (Arezzo, Loi and Zuddas in Math. Z. 275:1207–1216,2013) for the computation of the Szegö kernel of the disk bundle over an Hermitian symmetric space of compact type.

We study the functorially finite maximal rigid subcategories in 2-CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously functorially finite and maximal rigid; we prove that the converse is true if the 2-CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of Keller and Reiten (2007) [KR], we prove that any functorially finite maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via simple mutations, then their endomorphism algebras have the same representation type.

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