Abstract
In the present paper, we prove a rigidity theorem for complete submanifolds with parallel Gaussian mean curvature vector in the Euclidean space \({\mathbb {R}}^{n+p}\) under an integral curvature pinching condition, which is a unified generalization of some rigidity results for self-shrinkers and the \(\lambda \)-hypersurfaces in Euclidean spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cao, H.D., Li, H.Z.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Partial Differ. Equ. 46, 879–889 (2013)
Cao, S.J., Xu, H.W., Zhao, E.T.: Pinching theorems for self-shrinkers of higher codimension. Preprint. 2014. http://www.cms.zju.edu.cn/upload/file/20170320/1489994331903839.pdf
Cheng, Q.M., Ogata, S., Wei, G.: Rigidity theorems of \(\lambda \)-hypersurfaces. Commun. Anal. Geom. 24, 45–58 (2016)
Cheng, Q.M., Peng, Y.J.: Complete self-shrinkers of the mean curvature flow. Calc. Var. Partial Differ. Equ. 52, 497–506 (2015)
Cheng, Q.M., Wei, G.: A gap theorem of self-shrinkers. Trans. Am. Math. Soc. 367, 4895–4915 (2015)
Cheng, Q.M., Wei, G.: The Gauss image of \(\lambda \)-hypersurfaces and a Bernstein type problem. arXiv:1410.5302
Cheng, Q.M., Wei, G.: Complete \(\lambda \)-hypersurfaces of weighted volume-preserving mean curvature flow. arXiv:1403.3177
Cheng, X., Mejia, T., Zhou, D.T.: Stability and compactness for complete \(f\)-minimal surfaces. Trans. Am. Math. Soc. 367, 4041–4059 (2015)
Cheng, X., Mejia, T., Zhou, D.T.: Simons-type equation for \(f\)-minimal hypersurfaces and applications. J. Geom. Anal. 25, 2667–2686 (2015)
Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I; generic singularities. Ann. Math. 175, 755–833 (2012)
Ding, Q., Xin, Y.L.: The rigidity theorems of self-shrinkers. Trans. Am. Math. Soc. 366, 5067–5085 (2014)
Ding, Q., Xin, Y.L., Yang, L.: The rigidity theorems of self shrinkers via Gauss maps. Adv. Math. 303, 151–174 (2016)
Guang, Q.: Gap and rigidity theorems of \(\lambda \)-hypersurfaces, arXiv:1405.4871
Hoffman, D., Spruck, J.: Soblev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)
Ilmanen, T.: Singularities of mean curvature flow of surfaces. Preprint, 1995. https://people.math.ethz.ch/~ilmanen/papers/pub.html
Le, N.Q., Sesum, N.: Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Commun. Anal. Geom. 19, 633–659 (2011)
Li, A.M., Li, J.M.: An instrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math. 58, 582–594 (1992)
Li, H.Z., Wei, Y.: Classification and rigidity of self-shrinkers in the mean curvature flow. J. Math. Soc. Jpn. 66, 709–734 (2014)
Li, X.X., Chang, X.F.: A rigidity theorem of \(\xi \)-submanifolds in \({\mathbb{C}}^2\). Geom. Dedic. 185, 155–169 (2016)
Lin, H.Z.: Some rigidity theorems for self-shrinkers of the mean curvature flow. J. Korean Math. Soc. 53, 769–780 (2016)
Lin, J.M., Xia, C.Y.: Global pinching theorem for even dimensional minimal submanifolds in a unit sphere. Math. Z. 201, 381–389 (1989)
McGonagle, M., Ross, J.: The hyperplane is the only stable, smooth solution to the isoperimetric problem in Gaussian space. Geom. Dedic. 178, 277–296 (2015)
Ogata, S.: A global pinching theorem of complete \(\lambda \)-hypersurfaces. arXiv:1504.00789
Shiohama, K., Xu, H.W.: A general rigidity theorem for complete submanifolds. Nagoya Math. J. 150, 105–134 (1998)
Wang, H.J., Xu, H.W., Zhao, E.T.: Gap theorems for complete \(\lambda \)-hypersurfaces. Pac. J. Math. 288, 453–474 (2017)
White, B.: Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math. 488, 1–35 (1997)
Xia, C.Y., Wang, Q.L.: Gap theorems for minimal submanifolds of a hyperbolic space. J. Math. Anal. Appl. 436, 983–989 (2016)
Xu, H.W.: A rigidity theorem for submanifolds with parallel mean curvature in a sphere. Arch. Math. 61, 489–496 (1993)
Xu, H.W., Gu, J.R.: A general gap theorem for submanifolds with parallel mean curvature in \({\mathbb{R}}^{n+p}\). Commun. Anal. Geom. 15, 175–194 (2007)
Xu, H.W., Gu, J.R.: \(L^2\)-isolation phenomenon for complete surfaces arising from Yang-Mills theory. Lett. Math. Phys. 80, 115–126 (2007)
Yau, S.T.: Submanifolds with constant mean curvature I. Am. J. Math. 96, 346–366 (1974)
Yau, S.T.: Submanifolds with constant mean curvature II. Am. J. Math. 97, 76–100 (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research supported by the National Natural Science Foundation of China, Grant Nos. 11531012, 11371315, 11201416.
Rights and permissions
About this article
Cite this article
Wang, H., Xu, H. & Zhao, E. Submanifolds with parallel Gaussian mean curvature vector in Euclidean spaces. manuscripta math. 161, 439–465 (2020). https://doi.org/10.1007/s00229-019-01104-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-019-01104-1