Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A global correspondence between
CMC-surfaces in $S^3$ and pairs
of non-conformal harmonic maps into $S^2$

Authors: R. Aiyama, K. Akutagawa, R. Miyaoka and M. Umehara
Journal: Proc. Amer. Math. Soc. 128 (2000), 939-941
MSC (2000): Primary 53C42; Secondary 53A10
Published electronically: October 25, 1999
MathSciNet review: 1707134
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show there is a global correspondence between branched constant mean curvature (i.e. CMC-) immersions in $S^3/\{\pm 1\}$ and pairs of non-conformal harmonic maps into $S^2$ in the same associated family. Furthermore, we give two applications.

References [Enhancements On Off] (What's this?)

  • 1. R. Aiyama and K. Akutagawa, Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere, to appear in Tôhoku Math. J.
  • 2. A.I. Bobenko, Constant mean curvature surfaces and integrable equations, Russian Math. Survey 46 (1991), 1-45. MR 93b:53009
  • 3. A.I. Bobenko, Surfaces in terms of 2 by 2 matrices: Old and new integrable cases, Harmonic maps and integrable systems (Eds. P. Fordy and J. C. Wood), Vieweg, (1994), 83-127. MR 95m:58047
  • 4. N.J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), 627-710. MR 91d:58050
  • 5. D.A. Hoffman, Jr. and R. Osserman, On the Gauss map of surfaces in $\mathbf{R}^3$ and $\mathbf{R}^4$, Proc. London Math. Soc. 50 (1985), 27-56. MR 86f:58034
  • 6. K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), 89-99. MR 81c:53005b
  • 7. H.B. Lawson, Jr. and R.A. Tribuzy, On the mean curvature function for compact surfaces, J. Differential Geom. 16 (1981), 179-183. MR 83e:53060
  • 8. R. Miyaoka, The splitting and deformations of the generalized Gauss map of compact CMC surfaces, Tôhoku Math. J. 51 (1999), 35-53. CMP 99:08
  • 9. W. Seaman, On surfaces in $\mathbf{R}^4$, Proc. Amer. Math. Soc. 94 (1985), 467-470. MR 86m:53011
  • 10. M. Umehara and K. Yamada, A deformation of tori with constant mean curvature in $\mathbf{R}^3$ to those in other space forms, Trans. Amer. Math. Soc. 330 (1992), 845-857. MR 92f92f:53013

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C42, 53A10

Retrieve articles in all journals with MSC (2000): 53C42, 53A10

Additional Information

R. Aiyama
Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan

K. Akutagawa
Affiliation: Department of Mathematics, Shizuoka University, Shizuoka 422-8529, Japan

R. Miyaoka
Affiliation: Department of Mathematics, Sophia University, Tokyo 102-8554, Japan

M. Umehara
Affiliation: Department of Mathematics, Hiroshima University, Hiroshima 739-8526, Japan

Received by editor(s): April 15, 1998
Published electronically: October 25, 1999
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society