Skip to main content
SpringerLink
Log in

Volume Renormalization for the Blaschke Metric on Strictly Convex Domains

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We consider the volume expansion of the Blaschke metric, which is a projectively invariant metric on a strictly convex domain in a locally flat projective manifold. When the boundary is even dimensional, we express the logarithmic coefficient L as the integral of affine invariants over the boundary. We also formulate an intrinsic geometry of the boundary as a conformal Codazzi structure and show that L gives a global conformal invariant of the boundary.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albin, P.: Renormalizing curvature integrals on Poincaré-Einstein manifolds. Adv. Math. 221, 140–169 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, M.: \(L^2\)-curvature and renormalization of AHE metrics on 4-manifolds. Math. Res. Lett. 8, 171–188 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bailey, T.N., Eastwood, M.G., Gover, A.R.: Thomas’ structure bundles for conformal, projective and related structures. Rocky Mt. J. 24, 1191–1217 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burstall, F.E., Calderbank, D.M.J.: Submanifold geometry in generalized flag manifolds, in Winter School in Geometry and Physics (Srni, 2003). Rend. del Circ. mat. di Palermo 72, 13–41 (2004)

  5. Calderbank, D.M.J.: Möbius structures and two-dimensional Einstein-Weyl geometry. J. Reine Angew. Math. 504, 37–53 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Čap, A., Gover, A.R.: Tractor calculi for parabolic geometries. Trans. Am. Math. Soc. 354(4), 1511–1548 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Čap, A., Gover, A.R.: Projective compactness and conformal boundaries. Math. Ann. 366(3–4), 1587–1620 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Čap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2009)

    MATH  Google Scholar 

  9. Čap, A., Slovák, A., Souček, V.: Bernstein–Gelfund–Gelfund sequence. Ann. Math. (2) 154, 97–113 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Case, J., Yang, P.: A Paneitz-type operator for CR pluriharmonic functions. Bull. Inst. Math. Acad. Sin. 8, 285–322 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Cheng, S.Y., Yau, S.T.: On the regularity of the Monge–Ampère equation \(\text{ det }({\partial }^2u/{\partial } x^i{\partial } x^j)=F(x, u)\). Commun. Pure Appl. Math. 30, 41–68 (1977)

    Article  MATH  Google Scholar 

  12. Cheng, S.Y., Yau, S.T.: On the existence of a complete Kähler–Einstein metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math. 123, 507–544 (1980)

    Article  MATH  Google Scholar 

  13. Fefferman, C.: Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. 103, 395–416 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fefferman, C., Graham, C.R.: \(Q\)-curvature and Poincaré metric. Math. Res. Lett. 10, 819–832 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fox, D.J.F.: Geometric structures modeled on affine hypersurfaces and generalization of the Einstein Weyl and affine hypersphere equations. arXiv:0909.1897

  16. Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. Rend. Cirs. Mat. Palermo Ser. II 63(Suppl), 31–42 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guillarmou, C., Moroianu, S., Schlenker, J.-M.: The renormalized volume and uniformisation of conformal structures. arXiv:1211.6705

  19. Henningson, M., Skenderis, K.: The holographic Weyl anomaly. J. High Energy Phys. 7, 023 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Hirachi, K., Marugame, T., Matsumoto, Y.: Variations of total \(Q\)-prime curvature on CR manifolds. Adv. Math. 306, 1333–1376 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hirachi, K.: \(Q\)-prime curvature on CR manifolds. Differ. Geom. Appl. 33(Suppl), 213–245 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marugame, T.: GJMS operators and \(Q\)-curvature for conformal Codazzi structures. Differ. Geom. Appl. 49, 176–196 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nomizu, K., Sasaki, T.: Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  24. Randall, M.: The conformal-to-Einstein equation on Möbius surfaces. Differ. Geom. Appl. 35, 274–290 (2014)

    Article  MATH  Google Scholar 

  25. Sasaki, T.: A note on characteristic functions and projectively invariant metrics on a bounded convex domain. Tokyo J. Math. 8, 49–79 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sasaki, T.: On the characteristic function of a strictly convex domain and Fubini–Pick invariant. Results Math. 13, 367–378 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Seshadri, N.: Volume renormalization for complete Einstein–Kähler metrics. Differ. Geom. Appl. 25, 356–379 (2007)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This paper is based on part of the author’s thesis at the University of Tokyo. I would like to express my deep gratitude to my advisor Professor Kengo Hirachi for his support and encouragement throughout this work. I would also like to thank Dr. Yoshihiko Matsumoto, Professor Michael Eastwood, and Professor Bent Ørsted for invaluable comments on the results. This research was partially supported by the JSPS Fellowship and KAKENHI 13J06630.

Author information

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

    Taiji Marugame

Authors
  1. Taiji Marugame
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taiji Marugame.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marugame, T. Volume Renormalization for the Blaschke Metric on Strictly Convex Domains. J Geom Anal 28, 510–545 (2018). https://doi.org/10.1007/s12220-017-9831-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-017-9831-2

Keywords

Mathematics Subject Classification

Search

Navigation