On Holomorphic Isometric Embeddings of Complex Unit Balls Into Bounded Symmetric Domains

This dissertation, "On Holomorphic Isometric Embeddings of Complex Unit Balls Into Bounded Symmetric Domains" by Shan-tai, Sandy, Chan, 陳山大, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled ON HOLOMORPHIC ISOMETRIC EMBEDDINGS OF COMPLEX UNIT BALLS INTO BOUNDED SYMMETRIC DOMAINS Submitted by CHAN Shan Tai Sandy for the Degree of Doctor of Philosophy at The University of Hong Kong in September 2016 In 1953, Calabi [Ca53] studied (local) holomorphic isometries of K ahler man- ifolds into complex space forms and their local rigidity. On the other hand, the study of holomorphic isometries is also related to arithmetic geometry due to the work of Clozel-Ullmo [CU03] in which they studied modular correspon- dences which led to the study of local holomorphic isometries of an irreducible bounded symmetric domain (BSD) into its nite Cartesian power up to scalar constants. They further conjectured that for any positive integerq, any germ of holomorphic isometry from the unit disk into the q-disk with respect to their Bergman metrics up to normalizing constants is totally geodesic. Later on, Mok [Mk12] disproved this conjecture by constructing thep-th root embedding, which is not totally geodesic. In 2008, Ng [Ng08] developed a systematic way to study all holomorphic isometries ! with respect to their Bergman metrics up to normalizing constants by imposing conditions on the sheeting numbers to such isometries (cf. [Ng10]). Then, Ng [Ng08] proved that the p-th root embedding is globally rigid in the sense that any holomorphic isometry ! with the isometric constant equal to 1 and the global sheeting number equal to p, is the p-th root embedding up to reparametrizations when p 2 is an odd integer or p = 2. The case where p 4 is an even integer remains unknown until the research carried out by the author in this thesis.n In this thesis, holomorphic isometries of complex unit ballsB into bounded symmetric domains (BSDs) with respect to their Bergman metrics up to nor- malizing constants were studied. One of the key ingredients in the study is the polarized functional equation obtained from such holomorphic isometries (cf. [Ng08, Mk11, Mk12]). Firstly, we proved the global rigidity of the p-th root embedding for any integer p 2. After that, we classied all holomorphic isometries of into the 4-disk . In particular this provided an armative answer to Problem 5.1.2. in [Mk11, p. 262] when the target of such isometries is the 4-disk. In addition, we classied all holomorphic isometries ! with certain prescribed sheeting numbers. Secondly, holomorphic isometries of B into irreducible BSDs with respect to their Bergman metrics and with the minimal isometric constant were studied. It was discovered that such isometries arise from linear sections of minimal embeddings of the compact dual Hermitian symmetric manifolds of the BSDs by applying the general theory developed in [Mk12]. For the particular case where is an irreducible type-IV BSD, alias a Lie ball, i.e., a BSD dual to a hyperquadric, images of all holomorphic isome- tries ofB into were classied, and this study was generalized to the study of all holomorphic isometries ofB into certain irreducible BSDs of rank 2. Fi- nally, we studied holomorphic isometries ofB into an irreducible BSD with non-minimal isometric constants. In particular, results concerning the upper bound of the dimension of isometrically embeddedB in and the structure of the images of such isometries were obtained. Subjects: Geom