Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

TL;DR

This paper surveys recent advances in collapsing Riemannian manifolds with Ricci curvature bounds and proves that certain volume-collapsed Calabi-Yau manifolds admit Ricci-flat Kähler metrics with nilpotent symmetry structures.

Contribution

It introduces new results on Ricci-flat metrics and nilpotent structures on collapsed Calabi-Yau manifolds, addressing an open question in the field.

Findings

Collapsed Calabi-Yau manifolds admit Ricci-flat Kähler metrics

Existence of compatible nilpotent Killing structures on these manifolds

Advances in Ricci flow smoothing techniques for collapsing geometries

Abstract

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat K\"ahler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.