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

TitleSome rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains
Authors
Issue Date2017
PublisherThe 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.
AbstractThe 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.
DegreeDoctor of Philosophy
SubjectIsometrics (Mathematics)
Embeddings (Mathematics)
Geometry, Differential
Dept/ProgramMathematics
Persistent Identifierhttp://hdl.handle.net/10722/239980
HKU Library Item IDb5846391

 

DC FieldValueLanguage
dc.contributor.authorYang, Xiaoyu-
dc.contributor.author杨笑宇-
dc.date.accessioned2017-04-08T23:13:21Z-
dc.date.available2017-04-08T23:13:21Z-
dc.date.issued2017-
dc.identifier.citationYang, 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.urihttp://hdl.handle.net/10722/239980-
dc.description.abstractThe 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.languageeng-
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)-
dc.relation.ispartofHKU Theses Online (HKUTO)-
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.-
dc.rightsThis work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.-
dc.subject.lcshIsometrics (Mathematics)-
dc.subject.lcshEmbeddings (Mathematics)-
dc.subject.lcshGeometry, Differential-
dc.titleSome rigidity results about holomorphic isometric embeddings from complex unit balls into bounded symmetric domains-
dc.typePG_Thesis-
dc.identifier.hkulb5846391-
dc.description.thesisnameDoctor of Philosophy-
dc.description.thesislevelDoctoral-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-

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