A deformation of tori with constant mean curvature in to those in other space forms

Authors:
Masaaki Umehara and Kotaro Yamada

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that tori with constant mean curvature in constructed by Wente can be deformed to tori with constant mean curvature in the hyperbolic -space or the -sphere.

**[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*-*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*-*tori into the*-*sphere*, Geometry of Manifolds, Academic Press, 1989, pp. 151-161. MR**1040523 (91e:58051)****[5]**R. Walter,*Explicit examples to the*-*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*, Seminar on New Results in Nonlinear Partial Differential Equations, Max-Plank-Institut für Mathematik, Bonn, 1987, pp. 1-36.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53A10

Retrieve articles in all journals with MSC: 53A10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1050088-2

Keywords:
Surfaces with constant mean curvature

Article copyright:
© Copyright 1992
American Mathematical Society