Degenerations of Ricci-flat Calabi-Yau manifolds

This paper is a sequel to arXiv:1012.2940. We further investigate the Gromov-Hausdorff convergence of Ricci-flat K\"{a}hler metrics under degenerations of Calabi-Yau manifolds. We extend Theorem 1.1 in arXiv:1012.2940 by removing the condition on existence of crepant resolutions for Calabi-Yau varieties.Comment: An error is correcte

Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $\rho>0$ such that for any $x\in M$, the open $\rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_\rho,x^*)\to (B_\rho(x),x)$, has the maximal volume i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].Comment: The only change from the early version is an improvement on Theorem A: we replace the non-collapsing condition on $M$ by on $\tilde M$ (the Riemannian universal cover), and the corresponding modification is adding "subsection c" in Section

A New Proof of the Gromov's Theorem on Almost Flat Manifolds

We will present a new proof for the Gromov's theorem on almost flat manifolds ([Gr], [Ru]).Comment: 10 page