Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Geometric Complex Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 246))

  • 922 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anderson, M.T.: Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds. Adv. Math. 179, 205–249 (2003)

    Article  MathSciNet  Google Scholar 

  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. 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. Besse, A.L.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10. Springer-Verlag, Berlin (1987)

    Google Scholar 

  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. Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque (265), vi+109 (2000)

    Google Scholar 

  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. Wilson

    Google Scholar 

  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. Biquard, O., Mazzeo, R.: Parabolic geometries as conformal infinities of Einstein metrics. Arch. Math. (Brno) 42(5), 85–104 (2006)

    MathSciNet  MATH  Google Scholar 

  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. Bochner, S.: Analytic and meromorphic continuation by means of Green’s formula. Ann. Math. 2(44), 652–673 (1943)

    Article  MathSciNet  Google Scholar 

  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. 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. 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. Case, J.S., Gover, A.R.: The \(P^{\prime }\)-operator, the \(Q^{\prime }\)-curvature, and the CR tractor calculus (2017). arXiv:1709.08057

  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. 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. 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. Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)

    Article  MathSciNet  Google Scholar 

  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. 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. Fefferman, C., Graham, C.R.: The ambient metric. Ann. Math. Stud. 178 (2012). Princeton University Press, Princeton, NJ

    Google Scholar 

  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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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)

    Article  MathSciNet  Google Scholar 

  36. Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (1983). https://doi.org/10.1007/BF01393826

  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. 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. 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. Marugame, T.: Self-dual Einstein ACH metric and CR GJMS operators in dimension three (2018). arXiv:1802.01264

  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. Matsumoto, Y.: Asymptotically complex hyperbolic Einstein metrics and CR geometry (2013). The University of Tokyo

    Google Scholar 

  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. Matsumoto, Y.: Deformation of Einstein metrics and \(L^2\)-cohomology on strictly pseudoconvex domains (2016). arXiv:1603.02216

  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. 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. Takeuchi, Y.: \(Q\)-prime curvature and scattering theory on strictly pseudoconvex domains. Math. Res. Lett. 24(5), 1523–1554 (2017)

    Article  MathSciNet  Google Scholar 

  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. Takeuchi, Y.: On the renormalized volume of tubes over polarized Kähler-Einstein manifolds (2018). To appear in J. Geom. Anal

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Yoshihiko Matsumoto

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305-2125, USA

    Yoshihiko Matsumoto

Authors
  1. Yoshihiko Matsumoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshihiko Matsumoto .

Editor information

Editors and Affiliations

  1. Department of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  2. Department of Mathematics, Pusan National University, Busan, Korea (Republic of)

    Hong Rae Cho

  3. Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, Korea (Republic of)

    Sung Yeon Kim

  4. Department of Mathematics, Gyeongsang National University, Jinju, Korea (Republic of)

    Kang-Hyurk Lee

  5. Department of Mathematics, Kyung Hee University, Seoul, Korea (Republic of)

    Jong-Do Park

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Matsumoto, Y. (2018). Einstein Metrics on Strictly Pseudoconvex Domains from the Viewpoint of Bulk-Boundary Correspondence. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_18

Download citation

Publish with us

Policies and ethics

Search

Navigation