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)

 
 

 

Harmonic 2-spheres
with $r$ pairs of extra eigenfunctions


Author: Motoko Kotani
Journal: Proc. Amer. Math. Soc. 125 (1997), 2083-2092
MSC (1991): Primary 49F10; Secondary 58E20
DOI: https://doi.org/10.1090/S0002-9939-97-03771-4
MathSciNet review: 1372035
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the present paper, deformations of harmonic 2-spheres in the unit $n$-sphere $S^{n}$ respecting the degree are studied. The limit maps of such deformations are characterized as harmonic maps with extra eigenfunctions. The space $Harm_{d}(S^{2},S^{n})$ of harmonic 2-spheres in $S^{n}$ with fixed degree $d$ is described in terms of such deformations and the limit maps.


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

  • [B] J.L.M.Barbosa, On minimal immersions of $S^{2}$ into $S^{2m}(1)$, Trans.Amer.Math.Soc. 210 (1975), 75-105. MR 51:11362
  • [BW1] J.Bolton and L.M.Woodward, Moduli spaces of harmonic 2-spheres, Geometry and Topology of Submanifolds IV, World Scientific, 1992, pp. 143-151. MR 93g:58035
  • [BW2] J.Bolton and L.M.Woodward, The space of harmonic maps of $S^{2}$ into $S^{n}$, Geometry and Global Analysis, Report of the first MSJ International Research Institute, Tôhoku University, 1993, pp. 165-173. MR 96k:58053
  • [Ca] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geom. 1 (1976), 111-125. MR 38:1616
  • [Cr] T.A.Crawford, The space of harmonic maps from the 2-spheres to complex projective space, preprint, McGill University., 1993.
  • [E1] N.Ejiri, Minimal deformation of a nonfull minimal surface in $S^{4}(1)$, Compositio.Math. 90 (1994), 183-209. MR 95a:53098
  • [E2] N.Ejiri, The boundary of the space of full harmonic maps of $S^{2}$ into $S^{2m}(1)$ and extra eigenfunctions, in preparation.
  • [EK1] N.Ejiri and M.Kotani, Index and flat ends of minimal surfaces, Tokyo J. Math. 16 (1993), 37-48. MR 94g:53003
  • [EK2] N.Ejiri and M.Kotani, Minimal surfaces in $S^{2m}(1) $with extra eigenfunctions, Quart. J. Math. 43 (1992), 421-440. MR 93k:53061
  • [FGKO] M.Furuta, M.A.Guest , M.Kotani and Y.Ohnita, On the fundamental group of the space of harmonic 2-spheres in the n-sphere, Math. Z. 215 (1994), 503-518. MR 95e:58047
  • [GMO] M.A.Guest, M.Mukai and Y.Ohnita, On the topology of spaces of harmonic 2-spheres in symmetric spaces, in preparation.
  • [GO] M.A.Guest and Y.Ohnita, Group actions and deformation for harmonic maps, J. Math. Soc. Japan 45 (1993), 671-704. MR 94m:58058
  • [K] M.Kotani, Connectedness of the space of minimal 2-spheres in $S^{2m}(1)$, Proc. Amer. Math. Soc. 120 (1994), 803-810. MR 94e:58033
  • [L] B.Loo, The space of harmonic maps of $S^{2}$ into $S^{4}$, Trans. Amer. Math. Soc. 313 (1989), 81-103. MR 90k:58050
  • [M1] M.Mukai, On connectedness of the space of harmonic 2-spheres in quaternionic projective spaces, to appear, Tokyo J.Math..
  • [M2] M.Mukai, On connectedness of the space of harmonic 2-spheres in real Grassmann manifolds of 2-planes, Natur.Sci.Reo.Ochanomizu Univ. 44 (1993), 99-115. MR 95a:58031
  • [V] J.L.Verdier, Two dimensional $\sigma $ models and harmonic maps from $S^{2}$ to $S^{2n}$, Lecture Notes in Physics, vol. 180, Berlin, Springer, 1983, p. 136-141.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 49F10, 58E20

Retrieve articles in all journals with MSC (1991): 49F10, 58E20


Additional Information

Motoko Kotani
Affiliation: Department of Mathematics, Faculty of Sciences, Toho University, Funabashi, Chiba, 274, Japan
Email: kotani@tansei.cc.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-97-03771-4
Keywords: Harmonic 2-spheres, extra eigenfunctions, null curves
Received by editor(s): October 24, 1995
Received by editor(s) in revised form: February 1, 1996
Communicated by: Peter Li
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society