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Ngaiming Mok, HKU, Holomorphic isometries
of the complex unit ball into bounded symmetric domains Abstract In earlier works we proved that
any germ of holomorphic isometry f: D
→ D' (up to normalizing
constants) with respect to the Bergman metric between two bounded
domains extends to a proper holomorphic isometry
whenever the Bergman metrics are complete, and that moreover the graph of the
map extends to an affine-algebraic variety provided that the Bergman kernels
are rational functions. In particular this applies to bounded symmetric
domains in their Harish-Chandra realizations. We have also constructed
examples of nonstandard holomorphic isometries of
the Poincaré disk into certain bounded symmetric
domains. It has been unknown for some time whether the complex unit ball Bn, n ≥ 2, can be holomorphically
and isometrically embedded in a nonstandard way
into some bounded symmetricdomains. We
give a construction of such isometric embeddings
for n ≥ 2, and show some
examples where one can prove uniqueness up to automorphisms
of the domain and target spaces. We also explain the relevance of
holomorphic isometric embeddings of the complex
unit ball for the Hyperbolic Ax-Lindemann
Conjecture for arbitrary lattices in functional transcendence theory. |