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Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

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Abstract

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1

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References

  1. Osserman, R., A Survey of Minimal Surfaces, New York: Van Nostrand, 1969.

    MATH  Google Scholar 

  2. Hopf, H., Differential Geometry in Large, LNM 1000, Berlin: Springer-Verlag, 1983.

    Google Scholar 

  3. Pinkall, U., Sterling, I., On the classification of constant mean curvature tori, Ann. of Math, 1989, 130: 407–451.

    Article  MathSciNet  Google Scholar 

  4. Wente, H., Counterexample to a conjecture of H. Hopf, Pacific J. Math., 1986, 121: 193–243.

    MATH  MathSciNet  Google Scholar 

  5. Gu, C. H., Hu, H. S., Zhou, Zh. X., Darboux Transformation in Soliton Theory and Its Geometric Applications (in Chinese), Shanghai: Sci. & Tech. Pub., 1999.

    Google Scholar 

  6. Umehara, M., Yamada, K., Complete surfaces of constant mean curvature-1 in hyperbolic 3-space, Ann. of Math., 1993,137: 611–638.

    Article  MATH  MathSciNet  Google Scholar 

  7. Umehara, M., Yamada, K., A parametrization of Weierstrass formulae and perturbation of complete minimal surfaces in R3 into the hyperbolic space, J. Reine Angew Math., 1992, 432: 93–116.

    MATH  MathSciNet  Google Scholar 

  8. Babich, M, Bobenko, A., Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J., 1993, 72: 151–185.

    Article  MATH  MathSciNet  Google Scholar 

  9. Alekseevskij, D. V., Vinberg, E. B., Solodovnikov, A. S., Geometry of Space of Constant Curvature, in E. M. S., Vol. 29, Berlin: Springer-Verlag, 1993.

    Google Scholar 

  10. Lawson, B., Complete minimal surfaces in S3, Ann. of Math., 1970, 92: 335–374.

    Article  MathSciNet  Google Scholar 

  11. Bryant, R., Surfaces of mean curvature one in hyperbolic space, Asterisque, 1987, 154-155: 321-347.

  12. Bobenko, A., Surfaces in terms of 2 by 2 matrices: Old and new integrable cases, in Harmonic Maps and Integrable System (eds. Fordy, A., Wood, J.), Aspects of Mathematics, E, 1993, 23: 83-128.

  13. Dorfmerster, J., Pedit, F., Wu, H., Weierstrass type represention of harmonic maps into symmetric spaces, Comm. in Anal. and Geom., 1998, 6: 633–668.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, 230026, Hefei, China

    Qing Chen & Yi Cheng

Authors
  1. Qing Chen
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  2. Yi Cheng
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Chen, Q., Cheng, Y. Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation. Sci. China Ser. A-Math. 45, 1066–1075 (2002). https://doi.org/10.1007/BF02879990

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  • DOI: https://doi.org/10.1007/BF02879990

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