A characterization of compact surfaces with constant mean curvature
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- by Masaaki Umehara PDF
- Proc. Amer. Math. Soc. 108 (1990), 483-489 Request permission
Abstract:
Surfaces in a $3$-space of constant curvature, whose arbitrary sufficiently small open subsets admit a non-trivial isometric deformation preserving the mean curvature function, are called locally $H$-deformable. It is well known that surfaces with constant mean curvature which are not totally umbilical are all locally $H$-deformable. Conversely, we shall show in this paper that any compact locally $H$-deformable surface has constant mean curvature.References
- A. Gervasio Colares and Katsuei Kenmotsu, Isometric deformation of surfaces in $\textbf {R}^3$ preserving the mean curvature function, Pacific J. Math. 136 (1989), no. 1, 71–80. MR 971934
- H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
- H. Blaine Lawson Jr. and Renato de Azevedo Tribuzy, On the mean curvature function for compact surfaces, J. Differential Geometry 16 (1981), no. 2, 179–183. MR 638784
- Nicolaos Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. MR 1043269, DOI 10.2307/1971494
- Nicolaos Kapouleas, Constant mean curvature surfaces in Euclidean three-space, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 2, 318–320. MR 903742, DOI 10.1090/S0273-0979-1987-15575-3
- Renato de Azevedo Tribuzy, A characterization of tori with constant mean curvature in space form, Bol. Soc. Brasil. Mat. 11 (1980), no. 2, 259–274. MR 671469, DOI 10.1007/BF02584641
- Henry C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243. MR 815044
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 483-489
- MSC: Primary 53C42; Secondary 53A10
- DOI: https://doi.org/10.1090/S0002-9939-1990-0987616-2
- MathSciNet review: 987616