On the impossibility of four-dimensional complex-hyperbolic Einstein Dehn filling

Di Cerbo, Luca; Golla, Marco

doi:10.1090/proc/16086

On the impossibility of four-dimensional complex-hyperbolic Einstein Dehn filling
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by Luca F. Di Cerbo and Marco Golla

Proc. Amer. Math. Soc. 151 (2023), 281-294

DOI: https://doi.org/10.1090/proc/16086

Published electronically: August 12, 2022

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Abstract:

We show that the complex-hyperbolic Einstein Dehn filling compactification cannot possibly be performed in dimension four.

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, A survey of Einstein metrics on 4-manifolds, Handbook of geometric analysis, No. 3, Adv. Lect. Math. (ALM), vol. 14, Int. Press, Somerville, MA, 2010, pp. 1–39. MR 2743446
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
Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
Olivier Biquard, Policopié on differential geometry and global analysis, Available on the Author’s Webpage, 2007.
Benjamin Bakker and Jacob Tsimerman, The Kodaira dimension of complex hyperbolic manifolds with cusps, Compos. Math. 154 (2018), no. 3, 549–564. MR 3731257, DOI 10.1112/S0010437X1700762X
Jeff Cheeger and Mikhael Gromov, Bounds on the von Neumann dimension of $L^2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), no. 1, 1–34. MR 806699
Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128. MR 303460
Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
Luca Fabrizio Di Cerbo, Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications, Pacific J. Math. 255 (2012), no. 2, 305–315. MR 2928554, DOI 10.2140/pjm.2012.255.305
Gabriele Di Cerbo and Luca F. Di Cerbo, Effective results for complex hyperbolic manifolds, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 89–104. MR 3338610, DOI 10.1112/jlms/jdu065
Gabriele Di Cerbo and Luca F. Di Cerbo, On the canonical divisor of smooth toroidal compactifications, Math. Res. Lett. 24 (2017), no. 4, 1005–1022. MR 3723801, DOI 10.4310/MRL.2017.v24.n4.a4
Xianzhe Dai and Guofang Wei, Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds, Adv. Math. 214 (2007), no. 2, 551–570. MR 2349712, DOI 10.1016/j.aim.2007.02.010
Joel Fine and Bruno Premoselli, Examples of compact Einstein four-manifolds with negative curvature, J. Amer. Math. Soc. 33 (2020), no. 4, 991–1038. MR 4155218, DOI 10.1090/jams/944
G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. École Norm. Sup. (4) 4 (1971), 409–455. MR 309145, DOI 10.24033/asens.1217
F. Hirzebruch, Chern numbers of algebraic surfaces: an example, Math. Ann. 266 (1984), no. 3, 351–356. MR 730175, DOI 10.1007/BF01475584
Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435–441. MR 350657
Robion C. Kirby, The topology of $4$-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989. MR 1001966, DOI 10.1007/BFb0089031
Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. MR 2460872, DOI 10.2140/gt.2008.12.2587
Claude LeBrun, Four-dimensional Einstein manifolds, and beyond, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., vol. 6, Int. Press, Boston, MA, 1999, pp. 247–285. MR 1798613, DOI 10.4310/SDG.2001.v6.n1.a10
François Laudenbach and Valentin Poénaru, A note on $4$-dimensional handlebodies, Bull. Soc. Math. France 100 (1972), 337–344. MR 317343, DOI 10.24033/bsmf.1741
Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI 10.1016/0040-9383(85)90004-7
Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, Preprint, arXiv:math/0211159, 2002.
Grisha Perelman, Ricci flow with surgery on three-manifolds, Preprint, arXiv:math/0303109, 2003.
Carlo Petronio and Joan Porti, Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem, Expo. Math. 18 (2000), no. 1, 1–35. MR 1751141
Alexandre Preissmann, Quelques propriétés globales des espaces de Riemann, Comment. Math. Helv. 15 (1943), 175–216 (French). MR 10459, DOI 10.1007/BF02565638
John A. Thorpe, Some remarks on the Gauss-Bonnet integral, J. Math. Mech. 18 (1969), 779–786. MR 256307
William P. Thurston, The geometry and topology of three-manifolds, available at http://msri.org/publications/books/gt3m/, 1978.
G. Tian and S.-T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 574–628. MR 915840
Friedhelm Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, Invent. Math. 3 (1967), 308–333.
Friedhelm Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten II, Invent. Math. 4 (1968), 87–117.
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