Advertisement

Einstein Metrics on Strictly Pseudoconvex Domains from the Viewpoint of Bulk-Boundary Correspondence

  • Yoshihiko Matsumoto
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 246)

Abstract

We present an overview of the correspondence between asymptotically complex hyperbolic Einstein metrics and CR structures on the boundary at infinity, which is the complex version of that between Poincaré-Einstein metrics and conformal structures, with the main focus on existence results. We also propose several open problems.

Keywords

Einstein equation Strictly pseudoconvex domains CR structures 

Notes

Acknowledgements

I wish to express my gratitude to the hospitality of Stanford University, where I was working as a visiting member when the manuscript was written, and I am deeply grateful to Rafe Mazzeo for hosting the visit, for discussions, and for encouragements. I would also appreciate the careful reading of the manuscript by the reviewer. This work was partially supported by JSPS KAKENHI Grant Number JP17K14189 and JSPS Overseas Research Fellowship.

References

  1. 1.
    Anderson, M.T.: Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds. Adv. Math. 179, 205–249 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bailey, T.N., Eastwood, M.G., Graham, C.R.: Invariant theory for conformal and CR geometry. Ann. Math. 139(3), 491–552 (1994).  https://doi.org/10.2307/2118571
  3. 3.
    Bedford, E., Bell, S., Catlin, D.: Boundary behavior of proper holomorphic mappings. Mich. Math. J. 30(1), 107–111 (1983). http://projecteuclid.org/euclid.mmj/1029002793
  4. 4.
    Besse, A.L.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10. Springer-Verlag, Berlin (1987)Google Scholar
  5. 5.
    Biquard, O.: Einstein deformations of hyperbolic metrics. In: Surveys in Differential Geometry: Essays on Einstein Manifolds, Surveys in Differential Geometry, vol. 6, pp. 235–246. Int. Press, Boston, MA (1999).  https://doi.org/10.4310/SDG.2001.v6.n1.a9
  6. 6.
    Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque (265), vi+109 (2000)Google Scholar
  7. 7.
    Biquard, O.: Asymptotically symmetric Einstein metrics. SMF/AMS Texts and Monographs, vol. 13. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris (2006). Translated from the 2000 French original by Stephen S. WilsonGoogle Scholar
  8. 8.
    Biquard, O., Herzlich, M.: A Burns-Epstein invariant for ACHE 4-manifolds. Duke Math. J. 126(1), 53–100 (2005).  https://doi.org/10.1215/S0012-7094-04-12612-0
  9. 9.
    Biquard, O., Mazzeo, R.: Parabolic geometries as conformal infinities of Einstein metrics. Arch. Math. (Brno) 42(5), 85–104 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Biquard, O., Mazzeo, R.: A nonlinear Poisson transform for Einstein metrics on product spaces. J. Eur. Math. Soc. (JEMS) 13(5), 1423–1475 (2011).  https://doi.org/10.4171/JEMS/285
  11. 11.
    Bochner, S.: Analytic and meromorphic continuation by means of Green’s formula. Ann. Math. 2(44), 652–673 (1943)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Burns, D., Epstein, C.L.: Characteristic numbers of bounded domains. Acta Math. 164(1–2), 29–71 (1990).  https://doi.org/10.1007/BF02392751
  13. 13.
    Čap, A., Schichl, H.: Parabolic geometries and canonical Cartan connections. Hokkaido Math. J. 29(3), 453–505 (2000).  https://doi.org/10.14492/hokmj/1350912986
  14. 14.
    Čap, A., Slovák, J.: Parabolic geometries. I, Mathematical Surveys and Monographs, vol. 154. American Mathematical Society, Providence, RI (2009).  https://doi.org/10.1090/surv/154. Background and general theory
  15. 15.
    Case, J.S., Gover, A.R.: The \(P^{\prime }\)-operator, the \(Q^{\prime }\)-curvature, and the CR tractor calculus (2017). arXiv:1709.08057
  16. 16.
    Case, J.S., Yang, P.: A Paneitz-type operator for CR pluriharmonic functions. Bull. Inst. Math. Acad. Sin. (N.S.) 8(3), 285–322 (2013)Google Scholar
  17. 17.
    Cheng, S.Y., Yau, S.T.: On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Comm. Pure Appl. Math. 33(4), 507–544 (1980).  https://doi.org/10.1002/cpa.3160330404
  18. 18.
    Epstein, C.L., Melrose, R.B., Mendoza, G.A.: Resolvent of the Laplacian on strictly pseudoconvex domains. Acta Math. 167(1–2), 1–106 (1991).  https://doi.org/10.1007/BF02392446
  19. 19.
    Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fefferman, C.: Parabolic invariant theory in complex analysis. Adv. Math. 31(2), 131–262 (1979).  https://doi.org/10.1016/0001-8708(79)90025-2
  21. 21.
    Fefferman, C., Graham, C.R.: Conformal invariants. Astérisque Numero Hors Serie, pp. 95–116 (1985). The mathematical heritage of Élie Cartan (Lyon, 1984)Google Scholar
  22. 22.
    Fefferman, C., Graham, C.R.: The ambient metric. Ann. Math. Stud. 178 (2012). Princeton University Press, Princeton, NJGoogle Scholar
  23. 23.
    Fefferman, C., Hirachi, K.: Ambient metric construction of \(Q\)-curvature in conformal and CR geometries. Math. Res. Lett. 10(5–6), 819–831 (2003).  https://doi.org/10.4310/MRL.2003.v10.n6.a9
  24. 24.
    Fefferman, C.L.: Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. (2) 103(2), 395–416 (1976)Google Scholar
  25. 25.
    Gover, A.R., Graham, C.R.: CR invariant powers of the sub-Laplacian. J. Reine Angew. Math. 583, 1–27 (2005).  https://doi.org/10.1515/crll.2005.2005.583.1
  26. 26.
    Graham, C.R.: Higher asymptotics of the complex Monge-Ampère equation. Compos. Math. 64(2), 133–155 (1987). http://www.numdam.org/item?id=CM_1987__64_2_133_0
  27. 27.
    Graham, C.R.: Scalar boundary invariants and the Bergman kernel. In: Complex analysis, II (College Park, Md., 1985–86). Lecture Notes in Math., vol. 1276, pp. 108–135. Springer, Berlin (1987).  https://doi.org/10.1007/BFb0078958
  28. 28.
    Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87(2), 186–225 (1991).  https://doi.org/10.1016/0001-8708(91)90071-E
  29. 29.
    Gursky, M.J., Han, Q.: Non-existence of Poincaré-Einstein manifolds with prescribed conformal infinity. Geom. Funct. Anal. 27(4), 863–879 (2017).  https://doi.org/10.1007/s00039-017-0414-y
  30. 30.
    Hawking, S.W., Page, D.N.: Thermodynamics of black holes in anti-de Sitter space. Comm. Math. Phys. 87(4), 577–588 (1982/83). http://projecteuclid.org.stanford.idm.oclc.org/euclid.cmp/1103922135
  31. 31.
    Hirachi, K.: Construction of boundary invariants and the logarithmic singularity of the Bergman kernel. Ann. Math. (2) 151(1), 151–191 (2000).  https://doi.org/10.2307/121115
  32. 32.
    Hirachi, K.: \(Q\)-prime curvature on CR manifolds. Diff. Geom. Appl. 33(suppl.), 213–245 (2014).  https://doi.org/10.1016/j.difgeo.2013.10.013
  33. 33.
    Hirachi, K., Marugame, T., Matsumoto, Y.: Variation of total Q-prime curvature on CR manifolds. Adv. Math. 306, 1333–1376 (2017).  https://doi.org/10.1016/j.aim.2016.11.005
  34. 34.
    Hislop, P.D., Perry, P.A., Tang, S.H.: CR-invariants and the scattering operator for complex manifolds with boundary. Anal. PDE 1(2), 197–227 (2008).  https://doi.org/10.2140/apde.2008.1.197
  35. 35.
    Kohn, J.J., Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold. Ann. Math. 2(81), 451–472 (1965)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (1983).  https://doi.org/10.1007/BF01393826
  37. 37.
    Lee, J.M.: Fredholm operators and Einstein metrics on conformally compact manifolds. Mem. Am. Math. Soc. 183(864), vi+83 (2006).  https://doi.org/10.1090/memo/0864
  38. 38.
    Lee, J.M., Melrose, R.: Boundary behaviour of the complex Monge-Ampère equation. Acta Math. 148, 159–192 (1982).  https://doi.org/10.1007/BF02392727
  39. 39.
    Marugame, T.: Renormalized Chern-Gauss-Bonnet formula for complete Kähler-Einstein metrics. Am. J. Math. 138(4), 1067–1094 (2016).  https://doi.org/10.1353/ajm.2016.0034
  40. 40.
    Marugame, T.: Self-dual Einstein ACH metric and CR GJMS operators in dimension three (2018). arXiv:1802.01264
  41. 41.
    Marugame, T.: Some remarks on the total CR \(Q\) and \(Q^{\prime }\)-curvatures. SIGMA Symmetry Integr. Geom. Methods Appl. 14, 010, 8 p (2018).  https://doi.org/10.3842/SIGMA.2018.010
  42. 42.
    Matsumoto, Y.: Asymptotically complex hyperbolic Einstein metrics and CR geometry (2013). The University of TokyoGoogle Scholar
  43. 43.
    Matsumoto, Y.: Asymptotics of ACH-Einstein metrics. J. Geom. Anal. 24(4), 2135–2185 (2014).  https://doi.org/10.1007/s12220-013-9411-z
  44. 44.
    Matsumoto, Y.: Deformation of Einstein metrics and \(L^2\)-cohomology on strictly pseudoconvex domains (2016). arXiv:1603.02216
  45. 45.
    Matsumoto, Y.: GJMS operators, \(Q\)-curvature, and obstruction tensor of partially integrable CR manifolds. Diff. Geom. Appl. 45, 78–114 (2016).  https://doi.org/10.1016/j.difgeo.2016.01.002
  46. 46.
    Roth, J.C.: Perturbation of Kähler-Einstein metrics. ProQuest LLC, Ann Arbor, MI (1999). Ph.D. thesis. University of Washington. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9924131
  47. 47.
    Takeuchi, Y.: \(Q\)-prime curvature and scattering theory on strictly pseudoconvex domains. Math. Res. Lett. 24(5), 1523–1554 (2017)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Takeuchi, Y.: Ambient constructions for Sasakian \(\eta \)-Einstein manifolds. Adv. Math. 328, 82–111 (2018).  https://doi.org/10.1016/j.aim.2018.01.007
  49. 49.
    Takeuchi, Y.: On the renormalized volume of tubes over polarized Kähler-Einstein manifolds (2018). To appear in J. Geom. AnalGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations