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On the isolation phenomena of Einstein manifolds--submanifolds versions


Authors: Xiuxiu Cheng, Zejun Hu, An-Min Li and Haizhong Li
Journal: Proc. Amer. Math. Soc. 146 (2018), 1731-1740
MSC (2010): Primary 53C24; Secondary 53A15, 53C25, 53D12
DOI: https://doi.org/10.1090/proc/13901
Published electronically: January 12, 2018
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Abstract: In this paper, we study the isolation phenomena of Einstein manifolds from the viewpoint of submanifolds theory. First, for locally strongly convex Einstein affine hyperspheres we prove a rigidity theorem and as its direct consequence we establish a unified affine differential geometric characterization of the noncompact symmetric spaces $ \mathrm {E}_{6(-26)}/\mathrm {F}_4$ and $ \mathrm {SL}(m,\mathbb{R})/\mathrm {SO}(m)$, $ \mathrm {SL}(m,\mathbb{C})/\mathrm {SU}(m)$, $ \mathrm {SU}^*(2m)/\mathrm {Sp}(m)$ for each $ m\ge 3$. Second and analogously, for Einstein Lagrangian minimal submanifolds of the complex projective space $ \mathbb{C}P^n(4)$ with constant holomorphic sectional curvature $ 4$, we prove a similar rigidity theorem and as its direct consequence we establish a unified differential geometric characterization of the compact symmetric spaces $ \mathrm {E}_{6}/\mathrm {F}_4$ and $ \mathrm {SU}(m)/\mathrm {SO}(m)$, $ \mathrm {SU}(m)$, $ \mathrm {SU}(2m)/\mathrm {Sp}(m)$ for each $ m\ge 3$.


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Additional Information

Xiuxiu Cheng
Affiliation: School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
Email: chengxiuxiu1988@163.com

Zejun Hu
Affiliation: School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
Email: huzj@zzu.edu.cn

An-Min Li
Affiliation: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China
Email: anminliscu@126.com

Haizhong Li
Affiliation: Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People’s Republic of China
Email: hli@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/proc/13901
Keywords: Einstein manifold, symmetric space, affine hypersphere, Lagrangian submanifold, complex projective space.
Received by editor(s): January 5, 2017
Published electronically: January 12, 2018
Additional Notes: The first and second authors were supported by grants of NSFC-11371330 and 11771404, the third author was supported by grants of NSFC-11521061, and the fourth author was supported by grants of NSFC-11671224.
Communicated by: Lei Ni
Article copyright: © Copyright 2018 American Mathematical Society

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