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Examples of compact Einstein four-manifolds with negative curvature


Authors: Joel Fine and Bruno Premoselli
Journal: J. Amer. Math. Soc. 33 (2020), 991-1038
MSC (2010): Primary 53C21, 58J60
DOI: https://doi.org/10.1090/jams/944
Published electronically: September 14, 2020
MathSciNet review: 4155218
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Abstract: 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 four-manifolds $(X_k)$ previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12). The construction begins with a certain sequence $(M_k)$ of hyperbolic four-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.


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Additional Information

Joel Fine
Affiliation: Département de mathématiques, Université libre de Bruxelles, Belgium
MR Author ID: 761187
Email: joel.fine@ulb.ac.be

Bruno Premoselli
Affiliation: Département de mathématiques, Université libre de Bruxelles, Belgium
MR Author ID: 1052062
Email: bruno.premoselli@ulb.ac.be

Received by editor(s): February 3, 2018
Received by editor(s) in revised form: July 12, 2019, and November 4, 2019
Published electronically: September 14, 2020
Additional Notes: The first author was supported by ERC consolidator grant 646649 “SymplecticEinstein”. Both authors were supported by the FNRS grant MIS F.4522.15.
The second author was also the recipient of an FNRS chargé de recherche fellowship whilst this article was being written. Part of this research was carried out whilst the first author was a visitor at MSRI, and he thanks both them and the NSF (grant number DMS-1440140).
Article copyright: © Copyright 2020 American Mathematical Society