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Theorem 1 (Mok [Mo4]). Let λ > 0 and f : (D,λ dsD;x0)) → (Ω, dsΩ; y0) be a holomorphic isometry with respect to the Bergman metric up to a normalizing constant. Assume that the Bergman metrics on D and Ω are complete, that the Bergman kernel KD(z, w) on D extends to a rational function in (z, w), and that analogously the Bergman kernel KD(ξ, ζ) extends to a rational function in (ξ, ζ), Then, f extends to a proper holomorphic isometric embedding F : (D,λ dsD) → (Ω, dsΩ). Moreover, Graph(f) ⊂ D × Ω extends to an affine-algebraic subvariety V ⊂ C × C . In particular, the extension theorem above applies to the case where D b C and Ω b C are bounded symmetric domains in their Harish-Chandra realizations. When D is also irreducible and of rank ≥ 2, Clozel-Ullmo [CU] (2003) observed that the proof of Hermitian metric rigidity in Mok [Mo1,2] already yields the total geodesy of f . Thus, nonstandard, (i.e., not totally geodesic) holomorphic isometries from an irreducible bounded symmetric domain D into a (not necessarily irreducible) bounded symmetric domain Ω may exist only in the case where D is of rank 1, i.e., biholomorphic to the complex unit ball B of dimension n ≥ 1.

Holomorphic isometries of the complex unit ball into irreducible bounded symmetric domains