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New Construction for Spherical Minimal Immersions

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Abstract

We describe a general method of manufacturing new minimal immersions between round spheres out of old ones. The resulting spherical minimal immersions are given analytically in terms of the harmonic projection operator and have higher source dimensions. Applied to classical examples, this gives an abundance of new minimal immersions of even-dimensional spheres.

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References

  1. Calabi, E.: Minimal immersions of surfaces in euclidean spheres, J. Differential Geom. 1 (1967), 111–125.

    Google Scholar 

  2. DeTurck, D. and Ziller, W.: Minimal isometric immersions of spherical space forms in spheres, Comment. Math. Helv. 67 (1992), 428–458.

    Google Scholar 

  3. DoCarmo, M. and Wallach, N.: Minimal immersions of spheres into spheres, Ann. Math. 93 (1971), 43–62.

    Google Scholar 

  4. Escher, Ch.: Minimal isometric immersions of inhomogeneous space forms into spheres-a necessary condition for existence, Trans. Amer. Math. Soc.(1996) (to appear).

  5. Gauchman, H. and Toth, G.: Fine structure of the space of spherical minimal immersions, Trans. Amer. Math. Soc. 348(6) (1996), 2441–2463.

    Google Scholar 

  6. Mashimo, K.: Minimal immersions of 3-dimensional spheres into spheres, Osaka J. Math. 2 (1984), 721–732.

    Google Scholar 

  7. Mashimo, K.: Homogeneous totally real submanifolds in S 6, Tsukuba J. Math. 9 (1985), 185–202.

    Google Scholar 

  8. Muto, Y.: The space W 2 of isometric minimal immersions of the three-dimensional sphere into spheres, Tokyo J. Math. 7 (1984), 337–358.

    Google Scholar 

  9. Takahashi, T.: Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385.

    Google Scholar 

  10. Toth, G.: Harmonic Maps and Minimal Immersions through Representation Theory, Academic Press, Boston, 1990.

  11. Toth, G.: Eigenmaps and the space of minimal immersions between spheres, Indiana U. Math. J. 43(4) (1994).

  12. Toth, G. and Ziller, W.: Spherical minimal immersions of the 3-sphere, Preprint, 1996.

  13. Vilenkin, N. I.: Special Functions and the Theory of Group Representations, Transl. Math. Monographs 22, Amer. Math. Soc., Providence, 1968.

  14. Wallach, N: Minimal immersions of symmetric spaces into spheres, in Symmetric Spaces, Dekker, New York, 1972, pp. 1–40.

  15. Wang, M. and Ziller, W.: On isotropy irreducible Riemannian manifolds, Acta Math. 166 (1991), 223–261.

    Google Scholar 

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  1. Department of Mathematics, Rutgers University, Camden, NJ, 08102,m, U.S.A.

    Gabor Toth

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  1. Gabor Toth
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Toth, G. New Construction for Spherical Minimal Immersions. Geometriae Dedicata 67, 187–196 (1997). https://doi.org/10.1023/A:1004900218939

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  • DOI: https://doi.org/10.1023/A:1004900218939

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