JournalsjemsVol. 14, No. 5pp. 1617–1656

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

  • Ngaiming Mok

    University of Hong Kong, China
Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric cover
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Abstract

We study the extension problem for germs of holomorphic isometries f:(D;x0)(Ω;f(x0))f: (D;x_0) \to (\Omega;f(x_0)) up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics dsD2ds_D^2 on DD and dsΩ2ds_\Omega^2 on Ω\Omega. Our main focus is on boundary extension for pairs of bounded domains (D,Ω)(D,\Omega) such that the Bergman kernel KD(z,w)K_D(z,w) extends meromorphically in (z,w)(z,\overline w) to a neighborhood of D×D\overline D \times D, and such that the analogous statement holds true for the Bergman kernel KΩ(ζ,ξ)K_{\Omega}(\zeta,\xi) on Ω\Omega. Assuming that (D;dsD2)(D;ds_D^2) and (Ω;dsΩ2)(\Omega;ds_\Omega^2) are complete K\"ahler manifolds, we prove that the germ of map ff extends to a proper holomorphic isometric embedding such that Graph(f)(f) extends to a complex-analytic subvariety on some neigborhood of D×Ω\overline D \times \overline{\Omega}. In the event that the Bergman kernel KD(z,w)K_D(z,w) extends to a rational function in (z;w)(z;\overline w) and the analogue holds true for the Bergman kernel KΩ(ζ,ξ)K_{\Omega}(\zeta,\xi), we show that Graph(f)(f) extends to an affine-algebraic variety. Our results apply especially to pairs (D,Ω)(D,\Omega) of bounded symmetric domains in their Harish-Chandra realizations. When DD is the complex unit ball BnB^n of dimension n2n \ge 2, we obtain a new rigidity result which guarantees the total geodesy of the map under certain conditions. On the other hand, we construct examples of holomorphic isometries of the unit disk into polydisks which are not totally geodesic, answering in the negative a conjecture of Clozel-Ullmo's.

Cite this article

Ngaiming Mok, Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1617–1656

DOI 10.4171/JEMS/343