Abstract
In the present paper, the rigidity of hypersurfaces in Euclidean space is revisited. The Darboux equation is highlighted and two new proofs of the rigidity are given via energy method and maximal principle, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexandrov, A. D., On a class of closed surfaces, Mat. Sb., 4, 1938, 69–77.
Blaschke, W., Vorlesungen Uber Differentialgeometrie, 1, Julis Springer, Berlin, 1924.
Cohn-Vossen, E., Unstarre geochlossene Flachen, Math. Annalen, 102, 1930, 10–29.
Dajczer, M. and Rodriguez, L. L., Infinitesimal rigidity of Euclidean submanifolds, Ann. Inst. Four., 40(4), 1990, 939–949.
Dong, G. C., The semi-global isometric imbedding in R 3 of two dimensional Riemannian manifolds with Gaussian curvature changing sign clearly, J. Partial Diff. Eqs., 1, 1993, 62–79.
Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of the Second Order, Springer-Verlag, Berlin, 1998.
Guan, P. and Shen, X., A rigidity theorem for hypersurfaces in higher dimensional space forms, Contemporary Mathematics, 644, 2015, 61–65.
Guan, P., Wang, Z. and Zhang, X., A proof of the Alexanderov’s uniqueness theorem for convex surface in ℝ3, Annales de l’Institut Henri Poincaré Analyse non linéaire, 33, 2016, 329–336.
Han, Q. and Hong, J., Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs, 130, Amer. Math. Soc., Providence, RI, 2006.
Ivan Izmestiev, Infinitesimal rigidity of convex polyhedra through the second derivative of the Hilbert-Einstein functional, Canadian Journal of Mathematics, 66(4), 2014, 783–825
Kobayashi, S., Transformation Groups in Differential Geometry, Classics in Mathematics, 70, Springer-Verlag, Berlin, Heidelberg, 1995.
Li, C. H., Miao, P. Z. and Wang, Z. Z., Uniqueness of isometric immersions with the same mean curvature, J. Funct. Anal., https://doi.org/10.1016/j.jfa.2018.06.021.
Li, C. H. and Wang, Z. Z., The Weyl problem in warped product space, J. Diff. Geom., arXiv: 1603.01350v2.
Lin, C.-Y. and Wang, Y.-K., On isometric embeddings into anti-de sitter spacetimes, Int. Math. Res. Notices, 16, 2015, 7130–7161.
Yau, S. T., Lecture on Differential Geometry, in Berkeley, 1994.
Acknowledgements
The authors wish to thank Professor Pengfei Guan and Professor Zhizhang Wang for their valuable suggestions and comments. Part of the content also comes from Professor Wang’s contribution. The first author wishes to thank China Scholarship Council for its financial support. The first author also would like to thank McGill University for their hospitality.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This work was supported by the National Natural Science Foundation of China (No. 11871160) and the Fundamental Research Funds for the Central Universities of China (No. ZYGX2016J135).
Rights and permissions
About this article
Cite this article
Li, C., Xu, Y. The Rigidity of Hypersurfaces in Euclidean Space. Chin. Ann. Math. Ser. B 40, 439–456 (2019). https://doi.org/10.1007/s11401-019-0143-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-019-0143-7