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Transactions of the American Mathematical Society

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A deformation of tori with constant mean curvature in $ {\bf R}\sp 3$ to those in other space forms

Authors: Masaaki Umehara and Kotaro Yamada
Journal: Trans. Amer. Math. Soc. 330 (1992), 845-857
MSC: Primary 53A10
MathSciNet review: 1050088
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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.

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  • [1] U. Abresch, Constant mean curvature tori in terms of elliptic functions, J. Reine Angew. Math. 374 (1987), 169-192. MR 876223 (88e:53006)
  • [2] J. Spruck, The elliptic $ \sinh$-Gordon equation and the construction of toroidal soap bubbles, Lecture Notes in Math., vol. 1340, Springer-Verlag, 1988, pp. 275-381. MR 974618 (90i:35265)
  • [3] H. Tasaki, M. Umehara, and K. Yamada, Deformations of symmetric spaces of compact type to their noncompact duals, Japan. J. Math. 17 (1991) (to appear). MR 1145660 (92m:53089)
  • [4] M. Umehara and K. Yamada, Harmonic non-holomorphic maps of $ 2$-tori into the $ 2$-sphere, Geometry of Manifolds, Academic Press, 1989, pp. 151-161. MR 1040523 (91e:58051)
  • [5] R. Walter, Explicit examples to the $ H$-problem of Heinz Hopf, Geom. Dedicata 23 (1987), 187-213. MR 892400 (88i:53015a)
  • [6] -, Constant mean curvature tori with spherical curvature lines in noneuclidean geometry, Manuscripta Math. 63 (1989), 343-363. MR 986189 (90a:53016)
  • [7] H. C. Wente, Counterexamples to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), 193-243. MR 815044 (87d:53013)
  • [8] -, 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.

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Keywords: Surfaces with constant mean curvature
Article copyright: © Copyright 1992 American Mathematical Society

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