HOLOMORPHIC ISOMETRIES OF THE COMPLEX UNIT BALL INTO IRREDUCIBLE BOUNDED SYMMETRIC DOMAINS

Let Omega subset of S be an irreducible bounded symmetric domain of rank >= 2 embedded as an open subset in its dual Hermitian symmetric manifold S of the compact type. Write c(1)(S) = (p+2)delta, delta is an element of H-2(S,Z) congruent to Z being the positive generator. We prove that there exists a nonstandard holomorphic embedding of the (p + 1)-dimensional complex unit ball Bp+1 into Omega which is isometric with respect to canonical Kahler-Einstein metrics g resp. h normalized so that minimal disks are of constant Gaussian curvature -2. We construct such holomorphic isometries using varieties of minimal rational tangents (VMRTs). We also prove that n <= p + 1 for any holomorphic isometry f : (B-n, g) -> (Omega, h). Our proofs rely on an extension theorem for holomorphic isometries of Mok (2012), the asymptotic behavior due to Klembeck (1978) of standard complete Kahler metrics on strictly pseudoconvex domains, and the fine structure of boundaries of bounded symmetric domains in their Harish-Chandra realizations due to Wolf (1972).