A deformation of tori with constant mean curvature in $\textbf {R}^ 3$ to those in other space forms
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- by Masaaki Umehara and Kotaro Yamada PDF
- Trans. Amer. Math. Soc. 330 (1992), 845-857 Request permission
Abstract:
It is shown that tori with constant mean curvature in ${\mathbb {R}^3}$ constructed by Wente $[7]$ can be deformed to tori with constant mean curvature in the hyperbolic $3$-space or the $3$-sphere.References
- U. Abresch, Constant mean curvature tori in terms of elliptic functions, J. Reine Angew. Math. 374 (1987), 169–192. MR 876223, DOI 10.1515/crll.1987.374.169
- Joel Spruck, The elliptic sinh Gordon equation and the construction of toroidal soap bubbles, Calculus of variations and partial differential equations (Trento, 1986) Lecture Notes in Math., vol. 1340, Springer, Berlin, 1988, pp. 275–301. MR 974618, DOI 10.1007/BFb0082902
- Hiroyuki Tasaki, Masaaki Umehara, and Kotaro Yamada, Deformations of symmetric spaces of compact type to their noncompact duals, Japan. J. Math. (N.S.) 17 (1991), no. 2, 383–399. MR 1145660, DOI 10.4099/math1924.17.383
- Masaaki Umehara and Kotaro Yamada, Harmonic nonholomorphic maps of $2$-tori into the $2$-sphere, Geometry of manifolds (Matsumoto, 1988) Perspect. Math., vol. 8, Academic Press, Boston, MA, 1989, pp. 151–161. MR 1040523
- Rolf Walter, Explicit examples to the $H$-problem of Heinz Hopf, Geom. Dedicata 23 (1987), no. 2, 187–213. MR 892400, DOI 10.1007/BF00181275
- Rolf Walter, Constant mean curvature tori with spherical curvature lines in non-Euclidean geometry, Manuscripta Math. 63 (1989), no. 3, 343–363. MR 986189, DOI 10.1007/BF01168376
- Henry C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243. MR 815044, DOI 10.2140/pjm.1986.121.193 —, Twisted tori in constant mean curvature in ${{\mathbf {R}}^3}$, Seminar on New Results in Nonlinear Partial Differential Equations, Max-Plank-Institut für Mathematik, Bonn, 1987, pp. 1-36.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 330 (1992), 845-857
- MSC: Primary 53A10
- DOI: https://doi.org/10.1090/S0002-9947-1992-1050088-2
- MathSciNet review: 1050088