Universal embeddings of flag manifolds and rigidity phenomena

We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of Takeuchi [30] and Nakagawa-Takagi [27]. Using this embedding, we establish new rigidity phenomena for holomorphic isometries between homogeneous K\"ahler manifolds. As a first immediate consequence we show the triviality of a K\"ahler-Ricci soliton submanifod of $C \times \Omega$, where $C$ is a flag manifold and $\Omega$ is a homogeneous bounded domain. Secondly, we show that no \emph{weak-relative} relationship can occur among the fundamental classes of homogeneous K\"ahler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two K\"ahler manifolds are said to be \emph{weak relatives} if they share, up to local isometry, a common K\"ahler submanifold of complex dimension at least two. Our main result precisely shows that if $E$ is (possibly indefinite) flat, $C$ is a flag manifold, and $\Omega$ is a homogeneous bounded domain, then: $E$ is not weak relative to $C\times\Omega$; $C$ is not weak relative to $E\times\Omega$; $\Omega$ is not weak relative to $E\times C$. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from \emph{relatives} to the more flexible notion of \emph{weak relatives} and dispense with the earlier ''special'' restriction on the flag-manifold factor. This result also unifies previous rigidity results from the literature, e.g., [5, 6, 7, 9, 12, 13, 32].