Stability of Einstein metrics and effective hyperbolization in large Hempel distance

TL;DR

This paper proves that manifolds with nearly hyperbolic metrics admit nearby Einstein metrics, providing new insights into hyperbolization, Dehn filling, and drilling in 3-manifolds with large Hempel distance.

Contribution

It extends Tian's work by establishing the existence of Einstein metrics close to almost hyperbolic metrics, with applications to hyperbolization and Dehn filling in 3-manifolds.

Findings

Existence of Einstein metrics near almost hyperbolic metrics.

New proof of hyperbolization for 3-manifolds with large Hempel distance.

Analytic approach to Dehn filling and drilling of cusps and tubes.

Abstract

Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the $C^{2,\alpha}$-topology. In dimension $3$ the original manifold only needs to have finite volume, and the volume can be arbitrarily large. Applications include a new proof of the hyperbolization of $3$-manifolds of large Hempel distance yielding some new geometric control on the hyperbolic metric, and an analytic proof of Dehn filling and drilling that allows the filling and drilling of arbitrary many cusps and tubes.