## Examples of compact Einstein four-manifolds with negative curvature

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Joel Fine and Bruno Premoselli
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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.

## References

- Michael T. Anderson,
*Dehn filling and Einstein metrics in higher dimensions*, J. Differential Geom.**73**(2006), no. 2, 219–261. MR**2225518** - Michael T. Anderson,
*Geometric aspects of the AdS/CFT correspondence*, AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lect. Math. Theor. Phys., vol. 8, Eur. Math. Soc., Zürich, 2005, pp. 1–31. MR**2160865**, DOI 10.4171/013-1/1 - Thierry Aubin,
*Équations du type Monge-Ampère sur les variétés kählériennes compactes*, Bull. Sci. Math. (2)**102**(1978), no. 1, 63–95 (French, with English summary). MR**494932** - Thierry Aubin,
*Nonlinear analysis on manifolds. Monge-Ampère equations*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR**681859**, DOI 10.1007/978-1-4612-5734-9 - Richard H. Bamler,
*Construction of Einstein metrics by generalized Dehn filling*, J. Eur. Math. Soc. (JEMS)**14**(2012), no. 3, 887–909. MR**2911887**, DOI 10.4171/JEMS/321 - M. Berger,
*Les variétés Riemanniennes $(1/4)$-pincées*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**14**(1960), 161–170 (French). MR**140054** - Arthur L. Besse,
*Einstein manifolds*, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR**2371700** - G. Besson, G. Courtois, and S. Gallot,
*Entropies et rigidités des espaces localement symétriques de courbure strictement négative*, Geom. Funct. Anal.**5**(1995), no. 5, 731–799 (French). MR**1354289**, DOI 10.1007/BF01897050 - O. Biquard,
*Polycopié on differential geometry and global analysis*, Unpublished lecture notes, available at math.ens.fr/~biquard/dgga2007.pdf, 2007. - Olivier Biquard,
*Métriques d’Einstein asymptotiquement symétriques*, Astérisque**265**(2000), vi+109 (French, with English and French summaries). MR**1760319** - Simon Brendle and Richard Schoen,
*Manifolds with $1/4$-pinched curvature are space forms*, J. Amer. Math. Soc.**22**(2009), no. 1, 287–307. MR**2449060**, DOI 10.1090/S0894-0347-08-00613-9 - P. Buser and P. Sarnak,
*On the period matrix of a Riemann surface of large genus*, Invent. Math.**117**(1994), no. 1, 27–56. With an appendix by J. H. Conway and N. J. A. Sloane. MR**1269424**, DOI 10.1007/BF01232233 - Piotr T. Chruściel and Walter Simon,
*Towards the classification of static vacuum spacetimes with negative cosmological constant*, J. Math. Phys.**42**(2001), no. 4, 1779–1817. MR**1820431**, DOI 10.1063/1.1340869 - V. Cortés and A. Saha,
*Quarter-pinched Einstein metrics interpolating between real and complex hyperbolic metrics*, Math. Z.**290**(2018), no. 1-2, 155–166. MR**3848428**, DOI 10.1007/s00209-017-2013-x - Erwann Delay,
*Essential spectrum of the Lichnerowicz Laplacian on two tensors on asymptotically hyperbolic manifolds*, J. Geom. Phys.**43**(2002), no. 1, 33–44. MR**1911712**, DOI 10.1016/S0393-0440(02)00007-4 - G. Dolzmann and S. Müller,
*Estimates for Green’s matrices of elliptic systems by $L^p$ theory*, Manuscripta Math.**88**(1995), no. 2, 261–273. MR**1354111**, DOI 10.1007/BF02567822 - Joel Fine,
*Constant scalar curvature Kähler metrics on fibred complex surfaces*, J. Differential Geom.**68**(2004), no. 3, 397–432. MR**2144537** - Joel Fine, Kirill Krasnov, and Dmitri Panov,
*A gauge theoretic approach to Einstein 4-manifolds*, New York J. Math.**20**(2014), 293–323. MR**3193955** - Joel Fine and Dmitri Panov,
*Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold*, J. Differential Geom.**82**(2009), no. 1, 155–205. MR**2504773** - C. Robin Graham and John M. Lee,
*Einstein metrics with prescribed conformal infinity on the ball*, Adv. Math.**87**(1991), no. 2, 186–225. MR**1112625**, DOI 10.1016/0001-8708(91)90071-E - M. Gromov and W. Thurston,
*Pinching constants for hyperbolic manifolds*, Invent. Math.**89**(1987), no. 1, 1–12. MR**892185**, DOI 10.1007/BF01404671 - Larry Guth and Alexander Lubotzky,
*Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds*, J. Math. Phys.**55**(2014), no. 8, 082202, 13. MR**3390717**, DOI 10.1063/1.4891487 - Emmanuel Hebey,
*Compactness and stability for nonlinear elliptic equations*, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. MR**3235821**, DOI 10.4171/134 - F. Hirzebruch,
*The signature of ramified coverings*, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 253–265. MR**0258060** - Gary R. Jensen,
*Homogeneous Einstein spaces of dimension four*, J. Differential Geometry**3**(1969), 309–349. MR**261487** - Michael Kapovich,
*Convex projective structures on Gromov-Thurston manifolds*, Geom. Topol.**11**(2007), 1777–1830. MR**2350468**, DOI 10.2140/gt.2007.11.1777 - Mikhail G. Katz, Mary Schaps, and Uzi Vishne,
*Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups*, J. Differential Geom.**76**(2007), no. 3, 399–422. MR**2331526** - Wilhelm Klingenberg,
*Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung*, Comment. Math. Helv.**35**(1961), 47–54 (German). MR**139120**, DOI 10.1007/BF02567004 - Norihito Koiso,
*Nondeformability of Einstein metrics*, Osaka Math. J.**15**(1978), no. 2, 419–433. MR**504300** - Claude LeBrun,
*Einstein metrics and Mostow rigidity*, Math. Res. Lett.**2**(1995), no. 1, 1–8. MR**1312972**, DOI 10.4310/MRL.1995.v2.n1.a1 - John M. Lee,
*Fredholm operators and Einstein metrics on conformally compact manifolds*, Mem. Amer. Math. Soc.**183**(2006), no. 864, vi+83. MR**2252687**, DOI 10.1090/memo/0864 - Plinio G. P. Murillo,
*Systole of congruence coverings of arithmetic hyperbolic manifolds*, Groups Geom. Dyn.**13**(2019), no. 3, 1083–1102. With an appendix by Cayo Dória and Murillo. MR**4002226**, DOI 10.4171/GGD/515 - H. Pedersen,
*Einstein metrics, spinning top motions and monopoles*, Math. Ann.**274**(1986), no. 1, 35–59. MR**834105**, DOI 10.1007/BF01458016 - F. Robert,
*Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d’ordre deux*, Unpublished notes, available at http://www.iecl.univ-lorraine.fr/\~Frederic.Robert/ConstrucGreen.pdf, 2010. - W. P. Thurston,
*The geometry and topology of three-manifolds*, Lecture notes distributed by Princeton University, available at http://library.msri.org/books/gt3m/, 1980. - Peter Topping,
*Lectures on the Ricci flow*, London Mathematical Society Lecture Note Series, vol. 325, Cambridge University Press, Cambridge, 2006. MR**2265040**, DOI 10.1017/CBO9780511721465 - Shing Tung Yau,
*On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I*, Comm. Pure Appl. Math.**31**(1978), no. 3, 339–411. MR**480350**, DOI 10.1002/cpa.3160310304

## 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). - © Copyright 2020 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**33**(2020), 991-1038 - MSC (2010): Primary 53C21, 58J60
- DOI: https://doi.org/10.1090/jams/944
- MathSciNet review: 4155218