Examples of compact Einstein four-manifolds with negative curvature

10 Mar 2020
•
Fine Joel
•
Premoselli Bruno

We give new examples of compact, negatively curved Einstein manifolds of
dimension $4$. These are seemingly the first such examples which are not
locally homogeneous...Our metrics are carried by a sequence of 4-manifolds
$(X_k)$ previously considered by Gromov and Thurston. The construction begins
with a certain sequence $(M_k)$ of hyperbolic 4-manifolds, each containing a
totally geodesic surface $\Sigma_k$ which is nullhomologous and whose normal
injectivity radius tends to infinity with $k$. For a fixed choice of natural
number $l$, we consider the $l$-fold cover $X_k \to M_k$ branched along
$\Sigma_k$. We prove that for any choice of $l$ and all large enough $k$
(depending on $l$), $X_k$ carries an Einstein metric of negative sectional
curvature. The first step in the proof is to find an approximate Einstein
metric on $X_k$, which is done by interpolating between a model Einstein metric
near the branch locus and the pull-back of the hyperbolic metric from $M_k$. The second step in the proof is to perturb this to a genuine solution to
Einstein's equations, by a parameter dependent version of the inverse function
theorem. The analysis relies on a delicate bootstrap procedure based on $L^2$
coercivity estimates.(read more)