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

Abstract:

Let $\Omega \subset S$ be an irreducible bounded symmetric domain of rank $\ge 2$ embedded as an open subset in its dual Hermitian symmetric manifold $S$ of the compact type. Write $c_1(S) = (p+2)\delta$, $\delta \in H^2(S,\mathbb {Z}) \cong \mathbb {Z}$ being the positive generator. We prove that there exists a nonstandard holomorphic embedding of the $(p+1)$-dimensional complex unit ball $B^{p+1}$ into $\Omega$ which is isometric with respect to canonical Kähler-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 $n \le p+1$ for any holomorphic isometry $f: (B^n,g) \to (\Omega ,h)$. Our proofs rely on an extension theorem for holomorphic isometries of Mok (2012), the asymptotic behavior due to Klembeck (1978) of standard complete Kähler metrics on strictly pseudoconvex domains, and the fine structure of boundaries of bounded symmetric domains in their Harish-Chandra realizations due to Wolf (1972).