Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-06T09:11:12.522Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal immersions of S2 and ℝP2 into ℂPn with few higher order singularities

Published online by Cambridge University Press:  24 October 2008

J. Bolton ,
L. M. Woodward  and
L. Vrancken
J. Bolton
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE
L. M. Woodward
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE
L. Vrancken
Affiliation:
Department Wiskunde, Katholieke Universiteit Lenven, Celestijnenlaan 2008, B3001 Lenven, Belgium
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we extend ideas developed in 2, 4 to study certain minimal immersions of S2 and ℝP2 into ℂPn. Here S2 denotes the unit sphere in ℝ3 with its standard conformal structure and ℝP2 is S2 factored out by the antipodal map, while ℂPn denotes complex projective n-space equipped with the FubiniStudy metric of constant holomorphic sectional curvature 4. Since ℝPn with its standard metric of constant curvature 1 is included in ℂPn as a totally geodesic submanifold, this includes the case of minimal immersions into the unit sphere Sn(1) with its standard metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 93 - 101
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Barbosa, J. L.. On minimal immersions of S2 into S2n. Trans. Amer. Math. Soc. 210 (1975), 75106.Google Scholar
2Bolton, J., Jensen, G. R., Rigoli, M. and Woodward, L. M.. On conformal minimal immersions of S2 into ࠶Pn. Math. Ann. 279 (1988), 599620.CrossRefGoogle Scholar
3Bolton, J. and Woodward, L. M.. On immersions of surfaces into space forms. Soochow J. Math. 14 (1988), 1131.Google Scholar
4Bolton, J. and Woodward, L. M.. On the Simon conjecture for minimal immersions with S1- symmetry. Math. Z. 200 (1988), 111121.CrossRefGoogle Scholar
5Chern, S. S.. On the minimal immersions of the two-sphere in a space of constant curvature. In Problems in Analysis (Princeton University Press, 1970), pp. 2740.Google Scholar
6Din, A. M. and Zakrzewski, W. J.. General classical solutions in the ࠶Pn1 model. Nuclear Phys. B 174 (1980), 397406.CrossRefGoogle Scholar
7Eells, J. and Wood, J. C.. Harmonic maps from surfaces to complex projective spaces. Adv. in Math. 49 (1983), 217263.CrossRefGoogle Scholar
8Ejiri, N.. Equivariant minimal immersions of S2 into S2m(1). Trans. Amer. Math. Soc. 297 (1986), 105124.Google Scholar
9Glaser, V. and Stora, R.. Regular solutions of the ࠶Pn models and further generalisations. Preprint (CERN, 1980).Google Scholar
10Griffiths, P. and Harris, J.. Principles of Algebraic Geometry (Wiley, 1978).Google Scholar
11Wolfson, J. G.. Harmonic sequences and harmonic maps of surfaces into complex Grassman manifolds. J. Differential Geom. 27 (1988), 161178.CrossRefGoogle Scholar