## Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains

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## Abstract:

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler–Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of the vanishing of an $L^2$ cohomology and taking advantage of the asymptotic complex hyperbolicity of the Cheng–Yau metric, we establish the possibility of such a deformation when the dimension of the domain is larger than or equal to three.## References

- Bernd Ammann, Robert Lauter, and Victor Nistor,
*On the geometry of Riemannian manifolds with a Lie structure at infinity*, Int. J. Math. Math. Sci.**1-4**(2004), 161–193. MR**2038804**, DOI 10.1155/S0161171204212108 - Aldo Andreotti and Edoardo Vesentini,
*Carleman estimates for the Laplace-Beltrami equation on complex manifolds*, Inst. Hautes Études Sci. Publ. Math.**25**(1965), 81–130. MR**175148**, DOI 10.1007/BF02684398 - Bo Berndtsson,
*The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman*, Ann. Inst. Fourier (Grenoble)**46**(1996), no. 4, 1083–1094 (English, with English and French summaries). MR**1415958**, DOI 10.5802/aif.1541 - Arthur L. Besse,
*Einstein manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR**867684**, DOI 10.1007/978-3-540-74311-8 - Olivier Biquard,
*Métriques d’Einstein asymptotiquement symétriques*, Astérisque**265**(2000), vi+109 (French, with English and French summaries). MR**1760319** - Olivier Biquard and Rafe Mazzeo,
*Parabolic geometries as conformal infinities of Einstein metrics*, Arch. Math. (Brno)**42**(2006), no. suppl., 85–104. MR**2322401** - Olivier Biquard and Rafe Mazzeo,
*A nonlinear Poisson transform for Einstein metrics on product spaces*, J. Eur. Math. Soc. (JEMS)**13**(2011), no. 5, 1423–1475. MR**2825169**, DOI 10.4171/JEMS/285 - Andreas Čap and Hermann Schichl,
*Parabolic geometries and canonical Cartan connections*, Hokkaido Math. J.**29**(2000), no. 3, 453–505. MR**1795487**, DOI 10.14492/hokmj/1350912986 - Andreas Čap and Jan Slovák,
*Parabolic geometries. I*, Mathematical Surveys and Monographs, vol. 154, American Mathematical Society, Providence, RI, 2009. Background and general theory. MR**2532439**, DOI 10.1090/surv/154 - So-Chin Chen and Mei-Chi Shaw,
*Partial differential equations in several complex variables*, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR**1800297**, DOI 10.1090/amsip/019 - Shiu Yuen Cheng and Shing Tung Yau,
*On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation*, Comm. Pure Appl. Math.**33**(1980), no. 4, 507–544. MR**575736**, DOI 10.1002/cpa.3160330404 - Paul R. Chernoff,
*Essential self-adjointness of powers of generators of hyperbolic equations*, J. Functional Analysis**12**(1973), 401–414. MR**0369890**, DOI 10.1016/0022-1236(73)90003-7 - Harold Donnelly,
*$L_2$ cohomology of pseudoconvex domains with complete Kähler metric*, Michigan Math. J.**41**(1994), no. 3, 433–442. MR**1297700**, DOI 10.1307/mmj/1029005071 - Harold Donnelly and Charles Fefferman,
*$L^{2}$-cohomology and index theorem for the Bergman metric*, Ann. of Math. (2)**118**(1983), no. 3, 593–618. MR**727705**, DOI 10.2307/2006983 - Harold Donnelly and Frederico Xavier,
*On the differential form spectrum of negatively curved Riemannian manifolds*, Amer. J. Math.**106**(1984), no. 1, 169–185. MR**729759**, DOI 10.2307/2374434 - C. L. Epstein, R. B. Melrose, and G. A. Mendoza,
*Resolvent of the Laplacian on strictly pseudoconvex domains*, Acta Math.**167**(1991), no. 1-2, 1–106. MR**1111745**, DOI 10.1007/BF02392446 - Charles L. Fefferman,
*Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains*, Ann. of Math. (2)**103**(1976), no. 2, 395–416. MR**407320**, DOI 10.2307/1970945 - G. B. Folland and J. J. Kohn,
*The Neumann problem for the Cauchy-Riemann complex*, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR**0461588** - Matthew P. Gaffney,
*Hilbert space methods in the theory of harmonic integrals*, Trans. Amer. Math. Soc.**78**(1955), 426–444. MR**68888**, DOI 10.1090/S0002-9947-1955-0068888-1 - C. Robin Graham,
*Higher asymptotics of the complex Monge-Ampère equation*, Compositio Math.**64**(1987), no. 2, 133–155. MR**916479** - 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 - John W. Gray,
*Some global properties of contact structures*, Ann. of Math. (2)**69**(1959), 421–450. MR**112161**, DOI 10.2307/1970192 - M. Gromov,
*Kähler hyperbolicity and $L_2$-Hodge theory*, J. Differential Geom.**33**(1991), no. 1, 263–292. MR**1085144**, DOI 10.4310/jdg/1214446039 - Lars Hörmander,
*$L^{2}$ estimates and existence theorems for the $\bar \partial$ operator*, Acta Math.**113**(1965), 89–152. MR**179443**, DOI 10.1007/BF02391775 - Lars Hörmander,
*An introduction to complex analysis in several variables*, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR**1045639** - N. Koiso,
*Einstein metrics and complex structures*, Invent. Math.**73**(1983), no. 1, 71–106. MR**707349**, DOI 10.1007/BF01393826 - 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 - John M. Lee and Richard Melrose,
*Boundary behaviour of the complex Monge-Ampère equation*, Acta Math.**148**(1982), 159–192. MR**666109**, DOI 10.1007/BF02392727 - Yoshihiko Matsumoto,
*GJMS operators, $Q$-curvature, and obstruction tensor of partially integrable CR manifolds*, Differential Geom. Appl.**45**(2016), 78–114. MR**3457389**, DOI 10.1016/j.difgeo.2016.01.002 - Ngaiming Mok and Shing-Tung Yau,
*Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions*, The mathematical heritage of Henri Poincaré, Part 1 (Bloomington, Ind., 1980) Proc. Sympos. Pure Math., vol. 39, Amer. Math. Soc., Providence, RI, 1983, pp. 41–59. MR**720056** - Takeo Ohsawa,
*Applications of the $\overline \partial$ technique in $L^2$ Hodge theory on complete Kähler manifolds*, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 413–425. MR**1128559** - John Charles Roth,
*Perturbation of Kaehler-Einstein metrics*, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–University of Washington. MR**2698907** - Craig van Coevering,
*Kähler-Einstein metrics on strictly pseudoconvex domains*, Ann. Global Anal. Geom.**42**(2012), no. 3, 287–315. MR**2972615**, DOI 10.1007/s10455-012-9313-5

## Additional Information

**Yoshihiko Matsumoto**- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan
- MR Author ID: 1025212
- Email: matsumoto@math.sci.osaka-u.ac.jp
- Received by editor(s): August 20, 2019
- Received by editor(s) in revised form: December 17, 2019
- Published electronically: May 26, 2020
- Additional Notes: This work was partially supported by Grant-in-Aid for JSPS Fellows (14J11754).
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 5685-5705 - MSC (2010): Primary 53C25; Secondary 32L20, 32Q20, 32T15, 32V05
- DOI: https://doi.org/10.1090/tran/8102
- MathSciNet review: 4127889