Complete hypersurfaces with 𝑤-constant mean curvature in the unit spheres

Cheng, Qing-Ming; Wei, Guoxin

doi:10.1090/tran/9076

Complete hypersurfaces with $w$-constant mean curvature in the unit spheres
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by Qing-Ming Cheng and Guoxin Wei;

Trans. Amer. Math. Soc. 377 (2024), 887-904

DOI: https://doi.org/10.1090/tran/9076

Published electronically: November 20, 2023

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Abstract:

In this paper, we study $4$-dimensional complete hypersurfaces with $w$-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for $4$-dimensional complete hypersurfaces with $w$-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition.

References

João Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), no. 3, 339–353. MR 731682, DOI 10.1007/BF01215045
Shaoping Chang, On minimal hypersurfaces with constant scalar curvatures in $S^4$, J. Differential Geom. 37 (1993), no. 3, 523–534. MR 1217159
Qing-Ming Cheng, The rigidity of Clifford torus $S^1(\sqrt {1/n})\times S^{n-1}(\sqrt {(n-1)/n})$, Comment. Math. Helv. 71 (1996), no. 1, 60–69. MR 1371678, DOI 10.1007/BF02566409
Qing-Ming Cheng and Susumu Ishikawa, A characterization of the Clifford torus, Proc. Amer. Math. Soc. 127 (1999), no. 3, 819–828. MR 1636934, DOI 10.1090/S0002-9939-99-05088-1
Qing Ming Cheng and Hisao Nakagawa, Totally umbilic hypersurfaces, Hiroshima Math. J. 20 (1990), no. 1, 1–10. MR 1050421
Qing Ming Cheng and Qian Rong Wan, Hypersurfaces of space forms $M^4(c)$ with constant mean curvature, Geometry and global analysis (Sendai, 1993) Tohoku Univ., Sendai, 1993, pp. 437–442. MR 1361210
Hongcang Yang and Qing-Ming Cheng, Chern’s conjecture on minimal hypersurfaces, Math. Z. 227 (1998), no. 3, 377–390. MR 1612653, DOI 10.1007/PL00004382
S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York-Berlin, 1970, pp. 59–75. MR 273546
Sebastião C. de Almeida and Fabiano G. B. Brito, Closed $3$-dimensional hypersurfaces with constant mean curvature and constant scalar curvature, Duke Math. J. 61 (1990), no. 1, 195–206. MR 1068385, DOI 10.1215/S0012-7094-90-06109-5
Qintao Deng, Huiling Gu, and Qiaoyu Wei, Closed Willmore minimal hypersurfaces with constant scalar curvature in $\Bbb {S}^5(1)$ are isoparametric, Adv. Math. 314 (2017), 278–305. MR 3658718, DOI 10.1016/j.aim.2017.05.002
Qi Ding and Y. L. Xin, On Chern’s problem for rigidity of minimal hypersurfaces in the spheres, Adv. Math. 227 (2011), no. 1, 131–145. MR 2782189, DOI 10.1016/j.aim.2011.01.018
Jianquan Ge and Zizhou Tang, Chern conjecture and isoparametric hypersurfaces, Differential geometry, Adv. Lect. Math. (ALM), vol. 22, Int. Press, Somerville, MA, 2012, pp. 49–60. MR 3076049
Juan-Ru Gu, Hong-Wei Xu, Zhi-Yuan Xu, and En-Tao Zhao, A survey on rigidity problems in geometry and topology of submanifolds, Proceedings of the Sixth International Congress of Chinese Mathematicians. Vol. II, Adv. Lect. Math. (ALM), vol. 37, Int. Press, Somerville, MA, 2017, pp. 79–99. MR 3701929
Zhen Guo and Haizhong Li, A variational problem for submanifolds in a sphere, Monatsh. Math. 152 (2007), no. 4, 295–302. MR 2358299, DOI 10.1007/s00605-007-0476-2
H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
Hideki Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205–214. MR 215259, DOI 10.2969/jmsj/01920205
Chia-Kuei Peng and Chuu-Lian Terng, Minimal hypersurfaces of spheres with constant scalar curvature, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 177–198. MR 795235
Chia Kuei Peng and Chuu-Lian Terng, The scalar curvature of minimal hypersurfaces in spheres, Math. Ann. 266 (1983), no. 1, 105–113. MR 722930, DOI 10.1007/BF01458707
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
Young Jin Suh and Hae Young Yang, The scalar curvature of minimal hypersurfaces in a unit sphere, Commun. Contemp. Math. 9 (2007), no. 2, 183–200. MR 2313512, DOI 10.1142/S021919970700237X
Zizhou Tang and Wenjiao Yan, On the Chern conjecture for isoparametric hypersurfaces, Sci. China Math. 66 (2023), no. 1, 143–162. MR 4529031, DOI 10.1007/s11425-022-1967-4
Hong Wei Xu, A rigidity theorem for submanifolds with parallel mean curvature in a sphere, Arch. Math. (Basel) 61 (1993), no. 5, 489–496. MR 1241055, DOI 10.1007/BF01207549
Hongwei Xu and Zhiyuan Xu, On Chern’s conjecture for minimal hypersurfaces and rigidity of self-shrinkers, J. Funct. Anal. 273 (2017), no. 11, 3406–3425. MR 3706607, DOI 10.1016/j.jfa.2017.08.020
Hong Cang Yang and Qing Ming Cheng, A note on the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Chinese Sci. Bull. 36 (1991), no. 1, 1–6. MR 1137751
Hong Cang Yang and Qing Ming Cheng, An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Manuscripta Math. 84 (1994), no. 1, 89–100. MR 1283329, DOI 10.1007/BF02567445
Hongcang Yang and Qing-Ming Cheng, Chern’s conjecture on minimal hypersurfaces, Math. Z. 227 (1998), no. 3, 377–390. MR 1612653, DOI 10.1007/PL00004382
Shing Tung Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, No. 102, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR 645728
Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203