In this note, we prove some gap theorems of asymptotic volume ratio for Ricci nonnegative metrics, and gap theorems of volume for Einstein metrics.

In this note, we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold (M,g) must be flat if it is Einstein, i.e. Ric_g=λg for some real number λ. (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.

Petrunin proved that the integral of scalar curvature in a unit ball is bounded from above in terms of the dimension of the manifold and the lower bound of the sectional curvature. In this paper, we give an alternative proof for this result. The main difference between this proof and Petrunin’s original proof is that we construct a stratified finite covering and apply it directly to the given manifold, rather than arguing by contradiction for a sequence of manifolds, which satisfy some technical lifting properties.

We present a metric approach to the study of moduli spaces of metrics with certain curvature bounds and suitable other geometric constraints, and their compactifications. This is accompanied by a deeper discussion of related results, questions and problems in the realm of positive and positively pinched sectional as well as Ricci curvature. Regarding the latter two topics, we place special emphasis on corresponding works and contributions of Rong X and his collaborators.

We prove a conjecture of Petrunin and Tuschmann on the non-existence of asymptotically flat 4-manifolds asymptotic to the half plane. We also survey recent progress and questions concerning gravitational instantons, which serve as our motivation for studying this question.

In this paper, we show that a smooth 4-manifold diffeomorphic to a complex hypersurface in $\mathbb{C} \mathbb{P}^{3}$ of degree d≥5 can be decomposed as the union of d(d−4)² copies of 4-dimensional pair-of-pants and certain subsets of K3 surfaces.