Classification of Möbius minimal and Möbius isotropic hypersurfaces in $ \mathbb{S}^{5} $

Bangchao Yin; Shujie Zhai; Bangchao Yin; Shujie Zhai

doi:10.3934/math.2021489

AIMS Mathematics

2021, Volume 6, Issue 8: 8426-8452. doi: 10.3934/math.2021489

Previous Article Next Article

Research article

Classification of Möbius minimal and Möbius isotropic hypersurfaces in $ \mathbb{S}^{5} $

Bangchao Yin ,
Shujie Zhai ^,

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

Received: 11 February 2021 Accepted: 04 May 2021 Published: 01 June 2021
MSC : 53A30, 53B25, 53C40

In this paper, we will prove that a closed Möbius minimal and Möbius isotropic hypersurface without umbilic points in the unit sphere $ \mathbb{S}^{5} $ is Möbius equivalent to either the torus $ \mathbb{S}^{2}(\frac{1}{\sqrt{2}})\times\mathbb{S}^{2}(\frac{1}{\sqrt{2}})\rightarrow \mathbb{S}^{5} $ or the Cartan minimal hypersurface in $ \mathbb{S}^{5} $ with four distinct principal curvatures.
- Möbius minimal,
- Möbius isotropic,
- isoparametric hypersurface
Citation: Bangchao Yin, Shujie Zhai. Classification of Möbius minimal and Möbius isotropic hypersurfaces in $ \mathbb{S}^{5} $[J]. AIMS Mathematics, 2021, 6(8): 8426-8452. doi: 10.3934/math.2021489

Related Papers:

Abstract

In this paper, we will prove that a closed Möbius minimal and Möbius isotropic hypersurface without umbilic points in the unit sphere $ \mathbb{S}^{5} $ is Möbius equivalent to either the torus $ \mathbb{S}^{2}(\frac{1}{\sqrt{2}})\times\mathbb{S}^{2}(\frac{1}{\sqrt{2}})\rightarrow \mathbb{S}^{5} $ or the Cartan minimal hypersurface in $ \mathbb{S}^{5} $ with four distinct principal curvatures.

References

[1]	E. Cartan, Sur des familles remarquables d'hypersurfaces isoparam$\acute{e}$triques dans les espaces spheriques, Math. Z., 45 (1939), 335-367. doi: 10.1007/BF01580289
[2]	S. P. Chang, A closed hypersurface with constant scalar curvature and constant mean curvature in $\mathbb{S}^{4}$ is isoparametric, Comm. Anal. Geom., 1 (1993), 71–100. doi: 10.4310/CAG.1993.v1.n1.a4
[3]	S. P. Chang, On minimal hypersurfaces with constant scalar curvatures in $\mathbb{S}^{4}$, J. Differ. Geom., 37 (1993), 523–534.
[4]	Q. T. Deng, H. L. Gu, Q. Y. Wei, Closed Willmore minimal hypersurfaces with constant scalar curvature in $\mathbb{S}^{5}(1)$ are isoparametric, Adv. Math., 314 (2017), 278–305. doi: 10.1016/j.aim.2017.05.002
[5]	Z. Guo, H. Li, C. P. Wang, The second variational formula for Willmore submanifolds in $\mathbb{S}^{n}$, Results Math., 40 (2001), 205–225. doi: 10.1007/BF03322706
[6]	Z. Guo, J. B. Fang, L. M. Lin, Hypersurfaces with isotropic Blaschke tensor, J. Math. Soc. Japan, 63 (2011), 1155–1186.
[7]	Z. J. Hu, H. Li, Classification of hypersurfaces with parallel Möbius second fundamental form in $\mathbb{S}^{n+1}$, Sci. China Ser. A, 47 (2004), 417–430. doi: 10.1360/03ys0134
[8]	Z. J. Hu, H. Z. Li, C. P. Wang, Classification of Möbius isoparametric hypersurfaces in $\mathbb{S}^{5}$, Monatsh. Math., 151 (2007), 201–222. doi: 10.1007/s00605-006-0420-x
[9]	Z. J. Hu, S. J. Zhai, Möbius isoparametric hypersurfaces with three distinct principal curvatures, Ⅱ, Pac. J. Math., 249 (2011), 343–370. doi: 10.2140/pjm.2011.249.343
[10]	Z. J. Hu, S. J. Zhai, Submanifolds with parallel Möbius second fundamental form in the unit sphere, Results Math., 73 (2018), 1–46. doi: 10.1007/s00025-018-0773-1
[11]	H. Li, Willmore hypersurfaces in a sphere, Asian J. Math., 5 (2001), 365–377. doi: 10.4310/AJM.2001.v5.n2.a4
[12]	H. Li, H. L. Liu, C. P. Wang, G. S. Zhao, Möbius isoparametric hypersurfaces in $\mathbb{S}^{n+1}$ with two distinct principal curvatures, Acta Math. Sin. 18 (2002), 437–446.
[13]	T. Z. Li, J. Qing, C. P. Wang, Möbius curvature, Laguerre curvature and Dupin hypersurface, Adv. Math., 311 (2017), 249–294. doi: 10.1016/j.aim.2017.02.024
[14]	H. L. Liu, C. P. Wang, G. S. Zhao, Möbius isotropic submanifolds in $\mathbb{S}^{n}$, Tohoku Math. J., 53 (2001), 553–569.
[15]	T. Lusala, M. Scherfner, L. A. M. Sousa Jr, Closed minimal Willmore hypersurfaces of $\mathbb{S}^5(1)$ with constant scalar curvature, Asian J. Math., 9 (2005), 65–78. doi: 10.4310/AJM.2005.v9.n1.a6
[16]	C. K. Peng, C. L. Terng, The scalar curvature of minimal hypersurfaces in spheres, Math. Ann., 266 (1983), 105–113. doi: 10.1007/BF01458707
[17]	Z. Z. Tang, W. J. Yan, On the Chern conjecture for isoparametric hypersurfaces, arXiv preprint arXiv: 2001.10134, 2020.
[18]	C. P. Wang, Möbius geometry of submanifolds in $\mathbb{S}^{n}$, Manuscripta Math., 96 (1998), 517–534. doi: 10.1007/s002290050080

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)