On the structure of almost Einstein manifolds

TL;DR

This paper investigates the geometric structure of limit spaces derived from sequences of almost Einstein manifolds, extending known properties of Einstein manifolds to this broader class under certain conditions.

Contribution

It establishes that under non-collapsing conditions, the limit space of almost Einstein manifolds shares key properties with Einstein manifold limits, with applications to Kähler geometry.

Findings

Limit spaces retain properties of Einstein manifold limits.

Results apply to Kähler manifolds.

Provides a framework for understanding almost Einstein manifolds.

Abstract

In this paper, we study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the $L^1$-sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein manifolds. As applications, we can apply our structure results to study the properties of K\"ahler manifolds.