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postgraduate thesis: Some rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains
Title  Some rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains 

Authors  
Issue Date  2017 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Yang, X. [杨笑宇]. (2017). Some rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. 
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. 
Degree  Doctor of Philosophy 
Subject  Isometrics (Mathematics) Embeddings (Mathematics) Geometry, Differential 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/239980 
HKU Library Item ID  b5846391 
DC Field  Value  Language 

dc.contributor.author  Yang, Xiaoyu   
dc.contributor.author  杨笑宇   
dc.date.accessioned  20170408T23:13:21Z   
dc.date.available  20170408T23:13:21Z   
dc.date.issued  2017   
dc.identifier.citation  Yang, X. [杨笑宇]. (2017). Some rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR.   
dc.identifier.uri  http://hdl.handle.net/10722/239980   
dc.description.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.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.subject.lcsh  Isometrics (Mathematics)   
dc.subject.lcsh  Embeddings (Mathematics)   
dc.subject.lcsh  Geometry, Differential   
dc.title  Some rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains   
dc.type  PG_Thesis   
dc.identifier.hkul  b5846391   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   