Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T08:57:16.978Z Has data issue: false hasContentIssue false

Moduli space of branched superminimal immersions of a compact Riemann surface into S4

Published online by Cambridge University Press:  09 April 2009

Bonaventure Loo
Affiliation:
Department of Mathematics Lower Kent Ridge Road, National Univesity of Singapore, Singapore, 119260 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max {2g, g + 2}, such spaces have the structure of projectivzed fibre products and are path-connected quasi projective varieties of dimension 2dg + 4. This generalizes known results for spaces of harmonic 2-spheres in S4.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

References

[ACGH]Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of algebraic curves, vol. I (Springer, New York, 1985).CrossRefGoogle Scholar
[Br]Bryant, R., ‘Conformal and minimal immersions of compact surfaces into the 4-spheres’, J. Differential Geom. 17 (1982), 455473.CrossRefGoogle Scholar
[BR]Burstall, F. E. and Rawnsley, J. H., Twistor theory for Riemannian symmetric spaces, Lect. Notes Math. 1424 (Springer, Berlin, 1990).CrossRefGoogle Scholar
[C]Calabi, E., ‘Quelques applications de l'analyse complexe aux surfaces d'aire minima’, in: Topics in complex manifolds (ed. Rossi, H.) (Les Presses de l'Univ. de Montréal, Montréal, 1967) pp. 5981.Google Scholar
[ES]Eells, J. and Salamon, S., ‘Twistorial construction of harmonic maps of surfaces into fourmanifolds’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), 589640.Google Scholar
[F]Fulton, W., Intersection theory (Springer, New York, 1984).CrossRefGoogle Scholar
[H]Hartshorne, R., Algebraic geometry (Springer, New York, 1977).CrossRefGoogle Scholar
[KL]Kobak, P. Z. and Loo, B., ‘Moduli of quaternionic superminimal immersions of 2-spheres into quaternionic projective spaces’, Ann. Global Anal. Geom., to appear.Google Scholar
[La]Lawson, H. B., ‘Surfaces minimales et la construction de Calabi–Penrose’, (Séminaire Bourbaki, Exposé 624, 1984),Google Scholar
Astérisque 121–122 (1985), 197211.Google Scholar
[L]Loo, B., ‘The space of harmonic maps of S2 into S4’, Trans. Amer. Math. Soc. 313 (1989), 81102.Google Scholar
[LV1]Loo, B. and Vainsencher, I., ‘Limits of graphs’, Mat. Contemp. 6 (1994), 4159.Google Scholar
[LV2]Loo, B. and Vainsencher, I., ‘Limits of graphs II’, Technical report, in preparation.Google Scholar
[W1]Wahl, J., ‘The Jacobian algebra of a graded Gorenstein singularity’, Duke Math. J. 55 (1987), 843871.CrossRefGoogle Scholar
[W2]Wahl, J., ‘Gaussian maps on algebraic curves’, J. Differential Geom. 32 (1990), 7798.CrossRefGoogle Scholar