Effective stability of negatively curved Einstein metrics in dimensions at least $4$

TL;DR

This paper proves that certain negatively curved manifolds close to Einstein metrics can be deformed into genuine Einstein metrics with negative curvature, regardless of size or injectivity radius constraints.

Contribution

It establishes the stability of negatively curved Einstein metrics under almost-Einstein conditions without dependence on geometric bounds.

Findings

Existence of genuine Einstein metrics from almost-Einstein metrics

Pinching constant independent of diameter, volume, and injectivity radius

Extension to manifolds of dimension at least four

Abstract

We show that if a closed manifold of dimension at least four admits a negatively curved metric that is almost Einstein in a suitable sense, then it admits a genuine Einstein metric of negative sectional curvature. Importantly, the pinching constant measuring the almost-Einstein condition neither depends on an upper bound for the diameter or volume, nor on a lower bound for the injectivity radius.