Universal constraints on the range of eigenmaps and spherical minimal immersions

Toth, Gabor

doi:10.1090/S0002-9947-99-02252-7

Universal constraints on the range of eigenmaps and spherical minimal immersions

Author: Gabor Toth
Journal: Trans. Amer. Math. Soc. 351 (1999), 1423-1443
MSC (1991): Primary 53C42
DOI: https://doi.org/10.1090/S0002-9947-99-02252-7
MathSciNet review: 1487632
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space.

Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 233294
Dennis DeTurck and Wolfgang Ziller, Minimal isometric immersions of spherical space forms in spheres, Comment. Math. Helv. 67 (1992), no. 3, 428–458. MR 1171304, DOI https://doi.org/10.1007/BF02566512
Manfredo P. do Carmo and Nolan R. Wallach, Minimal immersions of spheres into spheres, Ann. of Math. (2) 93 (1971), 43–62. MR 278318, DOI https://doi.org/10.2307/1970752
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI https://doi.org/10.2307/2373037
Norio Ejiri, Totally real submanifolds in a $6$-sphere, Proc. Amer. Math. Soc. 83 (1981), no. 4, 759–763. MR 630028, DOI https://doi.org/10.1090/S0002-9939-1981-0630028-6
Christine M. Escher, Minimal isometric immersions of inhomogeneous spherical space forms into spheres—a necessary condition for existence, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3713–3732. MR 1370639, DOI https://doi.org/10.1090/S0002-9947-96-01694-7

Escher Ch., On inhomogeneous space forms, Preprint, Oregon State University, 1997.

Hillel Gauchman and Gabor Toth, Fine structure of the space of spherical minimal immersions, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2441–2463. MR 1348151, DOI https://doi.org/10.1090/S0002-9947-96-01588-7

Katsuya Mashimo, Minimal immersions of $3$-dimensional sphere into spheres, Osaka J. Math. 21 (1984), no. 4, 721–732. MR 765352

John Douglas Moore, Isometric immersions of space forms in space forms, Pacific J. Math. 40 (1972), 157–166. MR 305312

Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 198393, DOI https://doi.org/10.2969/jmsj/01840380

Gábor Tóth, Classification of quadratic harmonic maps of $S^3$ into spheres, Indiana Univ. Math. J. 36 (1987), no. 2, 231–239. MR 891772, DOI https://doi.org/10.1512/iumj.1987.36.36013

Gábor Tóth, Harmonic maps and minimal immersions through representation theory, Perspectives in Mathematics, vol. 12, Academic Press, Inc., Boston, MA, 1990. MR 1042100

Gabor Toth, Eigenmaps and the space of minimal immersions between spheres, Indiana Univ. Math. J. 46 (1997), no. 2, 637–658. MR 1481607, DOI https://doi.org/10.1512/iumj.1997.46.1975

Toth G.-Ziller W., Spherical minimal immersions of the $3$-sphere, Comment. Math. Helvetici (to appear).

N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. Translated from the Russian by V. N. Singh. MR 0229863

Nolan R. Wallach, Minimal immersions of symmetric spaces into spheres, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Dekker, New York, 1972, pp. 1–40. Pure and Appl. Math., Vol. 8. MR 0407774

Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740