Some rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains

Abstract

The topic about isometric embeddings between two Riemannian manifolds is classic. In particular, let $(D, \mathrm{d}s^2_D)$ and $(\Omega, \mathrm{d}s^2_{\Omega})$ be two bounded symmetric domains equipped with Bergman metrics, and $f:(D,\mathrm{d}s^2_{D})\to (\Omega, \mathrm{d}s^2_{\Omega})$ be a holomorphic isometric embedding with respect to Bergman metrics. When $D$ is irreducible and rank$(D)\ge2$, Clozel Laurent and Ullmo Emmanuel observed that the proof of Hermitian metric rigidity by Ngaiming Mok already implied the total geodesy of the map $f$. Therefore, nonstandard, i.e. not totally geodesic, holomorphic isometries from $(D, \mathrm{d}s^2_D)$ to $(\Omega, \mathrm{d}s^2_{\Omega})$ can exist only for the case rank$(D)=1$. Based on the notion of $Varieties\ of\ Minimal\ Rational\ Tangents$ (VMRT), Ngaiming Mok explicitly constructed a holomorphic isometric embedding $F:(\mathbb{B}^{p+1}, \mathrm{d}s^2_{\mathbb{B}^{p+1}})\to (\Omega, \mathrm{d}s^2_{\Omega})$, where $p$ is the dimension of the VMRT of $\SSS$ (the compact dual of $\Omega$) at one point. He also showed that $(p+1)$ is the maximal dimension of a complex unit ball that can be isometrically embedded into $(\Omega, \mathrm{d}s^2_{\Omega})$. In this thesis, the uniqueness of such isometric embeddings, i.e. $f:(\mathbb{B}^{p+1}, \mathrm{d}s^2_{\mathbb{B}^{p+1}})\to (\Omega, \mathrm{d}s^2_{\Omega})$, will be studied. In the first chapter, Duality Principle is established and it is the main tool to be used in the following two chapters. In Chapter 2 and Chapter 3, it is proved that if $\Omega$ is one of (1) $D^{I}_{2,n}, n\ge3$, (2) $D^{I}_{3,n}, n\ge3$, (3) $\Omega^{E_6}$, (4) $\Omega^{E_7}$, then the image $S:=f(\mathbb{B}^{p+1})$ has a specific geometric structure, called the vertex structure. In the last chapter, as a preparation for the reconstruction process, the method called parallel transport of the second fundamental form is introduced. After that, the submanifold $S$ will be recovered by means of adjunction process.