The collapsing geometry of almost Ricci-flat 4-manifolds

TL;DR

This paper studies the geometric limits of almost Ricci-flat 4-manifolds, showing that under certain conditions, the limit space exhibits semiflat Kähler geometry away from singularities.

Contribution

It demonstrates that 4-manifolds with Ricci curvature tending to zero and converging to a 2D space are locally semiflat Kähler outside singular regions.

Findings

Limit spaces are two-dimensional under certain conditions.

The limiting geometry is semiflat Kähler away from curvature blowup regions.

Provides conditions under which the geometric structure simplifies in the limit.

Abstract

We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space, with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region is semiflat Kaehler.