Equivariant Embeddings of Kälerian Symmetric Spaces
J. -H. Eschenburg, K. K. Santos, R. Tribuzy
TL;DR
The paper investigates equivariant immersions of symmetric spaces, with a focus on Kähler symmetric spaces and the parallel pluri-mean curvature (ppmc) condition. Using the Cartan–Wallach decomposition of $C^{\infty}(P)$ into class-one $G$-modules, it shows that all equivariant embeddings arise as orbits of fundamental $f_k$ or finite sums thereof on compact spaces, and specializes to ${\mathbb{CP}}^n$ to perform explicit computations. By analyzing the normal space decompositions $N^+$, $N^-$ and, for Kähler spaces, $N^{++}$ and the $(1,1)$-part of the second fundamental form, it reduces the ppmc condition to a test on the fundamental embeddings. The main result is that among all equivariant embeddings of ${\mathbb{CP}}^n$ only the standard embedding (corresponding to $k=1$) has parallel pluri-mean curvature, establishing a rigidity phenomenon for ppmc in the equivariant setting.
Abstract
In this article we investigate some properties of equivariant embeddings of a symmetric Kählerian manifold. Motivated by a theorem of Cartan and Wallach on equivariant embeddings of symmetric spaces we characterize these embeddings in the special case of $\mathbb{CP}^n$. Further, we verify that if a equivariant embedding has parallel plurimean curvature then it is the extrinsically symmetric one.