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

TL;DR

This paper constructs a new example of a simply connected, nonspin 5-manifold that is almost Ricci-flat, collapsing with bounded sectional curvature, using a Kummer-type method, thus advancing understanding of Ricci-flat geometry.

Contribution

It introduces a novel Kummer-type construction of a simply connected, nonspin 5-manifold that is almost Ricci-flat and collapses with bounded sectional curvature.

Findings

Constructed a nonspin 5-manifold with almost Ricci-flat metrics.

Demonstrated the manifold collapses with bounded sectional curvature.

Provided new insights into Ricci-flat geometry in higher dimensions.

Abstract

A smooth closed manifold $M$ is called almost Ricci-flat if $$\inf_g||\textrm{Ric}_g||_\infty\cdot \textrm{diam}_g(M)^2=0$$ where $\textrm{Ric}_g$ and $\textrm{diam}_g$ denote the Ricci tensor and the diameter of $g$ respectively 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's minimal volume vanishes, namely it collapses with sectional curvature bounded.