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On the Szegő Metric

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Abstract

We introduce a new biholomorphically invariant metric based on Fefferman’s invariant Szegő kernel and investigate the relation of the new metric to the Bergman and Carathéodory metrics. A key tool is a new absolutely invariant function assembled from the Szegő and Bergman kernels.

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References

  1. Barrett, D.: A floating body approach to Fefferman’s hypersurface measure. Math. Scand. 98, 69–80 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Bell, S.: The Cauchy Transform, Potential Theory, and Conformal Mapping. CRC Press, Boca Raton (1992)

    Google Scholar 

  3. Burbea, J.: Effective methods of determining the modulus of doubly connected domains. J. Math. Anal. Appl. 62, 236–242 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Burns, D., Shnider, S.: Spherical hypersurfaces in complex manifolds. Invent. Math. 33, 223–246 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  5. Burns, D., Shnider, S.: Real hypersurfaces in complex manifolds. In: Proc. Sympos. Pure Math, vol. XXX, Part 2, pp. 141–168. Am. Math. Soc., Providence (1977)

    Google Scholar 

  6. Cheng, S.Y., Yau, S.T.: On the existence of a complex Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math. 33, 507–544 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chern, S., Ji, S.: On the Riemann mapping theorem. Ann. Math. 144(2), 421–439 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chern, S.-S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Fefferman, C.: Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. 103(2), 395–416 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fefferman, C.: Parabolic invariant theory in complex analysis. Adv. Math. 31, 131–262 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Graham, C.R.: Scalar boundary invariants and the Bergman kernel. In: Lecture Notes in Mathematics, vol. 1276, pp. 108–135. Springer, Berlin (1987)

    Google Scholar 

  13. Hahn, K.T.: Inequality between the Bergman metric and Carathéodory differential metric. Proc. Am. Math. Soc. 68, 193–194 (1978)

    MATH  Google Scholar 

  14. Hirachi, K.: A link between the asymptotic expansions of the Bergman kernel and the Szegő kernel. Adv. Stud. Pure Math. 42, 115–121 (2004)

    MathSciNet  Google Scholar 

  15. Hirachi, K., Komatsu, G., Nakazawa, N.: Two methods of determining local invariants in the Szegő kernel. In: Lecture Notes in Pure and Appl. Math., vol. 143, pp. 77–96. Dekker, New York (1993)

    Google Scholar 

  16. Lee, J., Melrose, R.: Boundary behavior of the complex Monge-Ampère equation. Acta Math. 148, 159–192 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pflug, P., Jarnicki, M.: Invariant Distances and Metrics in Complex Analysis. de Gruyter, Berlin (1993)

    MATH  Google Scholar 

  18. Pinčuk, S.I.: The analytic continuation of holomorphic mappings. Mat. Sb. 98, 416–435 (1975) (Russian). English trans. Math USSR Sb. 27, 375–392 (1975)

    MathSciNet  Google Scholar 

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Correspondence to Lina Lee.

Additional information

Communicated by Marco Abate.

The first author was supported in part by NSF grant number DMS-0901205.

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Barrett, D., Lee, L. On the Szegő Metric. J Geom Anal 24, 104–117 (2014). https://doi.org/10.1007/s12220-012-9329-x

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  • DOI: https://doi.org/10.1007/s12220-012-9329-x

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