Kummer-type constructions of almost Ricci-flat 5-manifolds

Abstract

A smooth closed manifold M is called almost Ricci-flat if

$$\begin{aligned} \inf _g||\text {Ric}_g||_\infty \cdot \text {diam}_g(M)^2=0 \end{aligned}$$

where \(\text {Ric}_g\) and \(\text {diam}_g\), respectively, denote the Ricci tensor and the diameter of g and g runs over all Riemannian metrics on M. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold M which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.