The sources of our investigation are the author’s constructions of nonstandard bona fide holomorphic isometric embeddings called p-th root maps of the Poincaré upper half-plane into Cartesian products of identical copies of itself, and those of complex unit balls \(\mathbb{B}^{p(\Omega)+1}\) of maximal admissible dimension into irreducible bounded symmetric domains Ω of rank ≥ 2 whose images are full cones Vb of minimal disks, b ∈ Reg(∂Ω). In this article we give a proof of a duality principle for holomorphic isometric embeddings of the complex unit ball \(\mathbb{B}^{m}\) , m ≥ 1 into an irreducible bounded symmetric domain Ω. Identify Ω by means of the Borel embedding as an open subset of its dual Hermitian symmetric space \(\widehat{\Omega}\) of the compact type. Let \(f:(\mathbb{B}^{m},g_{m})\rightarrow(\Omega, h)\) be a holomorphic isometry, where gm resp. h denotes the canonical Kähler–Einstein metric on \(\mathbb{B}^{m}\) resp. Ω normalized so that minimal disks are of constant Gaussian curvature −2, and assume \(0\in f(\mathbb{B}^{m})=:Z\) . Choosing a dual pair (G0, Gc) of noncompact resp. compact real forms of the connected simple complex Lie group G which is the identity component of \(\text{Aut}(\widehat{\Omega})\) so that G0 ∩ Gc = K is the isotropy subgroup of G0 at 0 ∈ Ω, and denoting by g the Gc-invariant Kähler–Einstein metric normalized so that minimal rational curves are of constant Gaussian curvature +2, the curvature tensors of (Ω, h) and \((\widehat{\Omega},g)\) at \(0 \in \Omega \subset\widehat{\Omega}\) are opposite to each other. Basing on the latter fact and comparing the two holomorphic isometric embeddings \(f:\mathbb{B}^{m}\rightarrow\Omega\) resp. \(\nu:\widehat{\Omega}\rightarrow\mathbb{P}^{N}\) , where the latter denotes the minimal projective embedding of \(\widehat{\Omega}\) , we obtain an identity on the lengths of the two second fundamental forms σ resp. τ applied to a pair of holomorphic tangent vectors ξ, η ∈ T0(Ω), and we call this the duality principle for holomorphic isometric embeddings of the complex unit ball into irreducible bounded symmetric domains. This duality principle yields constraints on isomorphism types of holomorphic tangent spaces of holomorphic isometric images Z ⊂ Ω of the complex unit ball, and it serves as the basis of uniqueness results on nonstandard holomorphic isometric embeddings \(f:\mathbb{B}^{p(\Omega)+1}\rightarrow{\Omega}\) of complex unit balls of maximal admissible dimension into Ω. By varying the base points we also amplify the duality principle to an identity on second fundamental forms σ and τ on Z. On the basis of the duality principle and other known restrictions which we collect as background information, we pose the question as to the validity of the aforementioned uniqueness statement for irreducible bounded symmetric domains of rank ɥ 2 other than the case of type-IV domains, for which uniqueness is known to fail. We also pose the question whether bona fide holomorphic isometric embeddings of the Poincaré upper half-plane into Cartesian products of identical copies of itself can always be reconstructed from applying various p-th root maps to Cartesian factors and from reparametrization.
