Consider a compact K¨ahler manifold Mm with Ricci curvature lower bound RicM ≥ −2 (m + 1) . Assume that its universal cover fM has maximal bottom of spectrum 1 fM = m2. Then we prove that fM is isometric to the complex hyperbolic space CHm.

A foliation F on a Riemannian manifold M is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold of M that intersects each leaf of F orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space M of noncompact type. These foliations are constructed as follows. Let  be an orthogonal subset of a set of simple roots associated with the symmetric space M. Then  determines a horospherical decomposition M = Fs ×ErankM−||×N, where Fs  is the Riemannian product of || symmetric spaces of rank one. Every hyperpolar homogeneous foliation on M is isometrically congruent to the product of the following objects: a particular homogeneous codimension one foliation on each symmetric space of rank one in Fs , a foliation by parallel affine subspaces on the Euclidean space ErankM−||, and the horocycle subgroup N of the parabolic subgroup of the isometry group of M determined by .

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point p in a spacetime N , we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at p . For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.

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In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Khler Calabi-Yau surfaces or Hopf surfaces.