Let D be a Hodge domain of dimension \(\ge 2\) canonically identified as an open subset of its compact dual rational homogeneous manifold Z presented as G/Q in standard notation and assumed throughout the article to be of Picard number 1. We show that there is a horizontal proper holomorphic isometric embedding of a complex unit ball \(\mathbb B^m\) , \(m \ge 2\) , into D coming from a cone of minimal rational curves. Except when D itself is a complex unit ball, such an isometry is not totally geodesic. In the long-root cases, i.e., the cases where \(Z = G/Q\) corresponds to the marking of the Dynkin diagram of \(\mathfrak g =\mathrm{Lie\,}G\) at a long simple root, the image of the holomorphic isometry is the nonempty intersection with D of the full cone \(\mathcal V_a\) of minimal rational curves emanating from some boundary point \(a \in \partial D\) , and we have \(m = p+1\) , where p is the dimension of the VMRT (variety of minimal rational tangents) \(\mathscr {C}_x(Z)\) at any point \(x \in Z\) . In the short-root cases, the image of the holomorphic isometry is the nonempty intersection with D of the cone of all minimal rational curves emanating from some boundary point \(a \in \partial D\) and tangent to the minimal invariant holomorphic distribution of Z.
