Conformal minimal two-spheres in Q ₂

Abstract

In this paper, we consider a conformal minimal immersion f from S ₂ into a hyperquadric Q ₂, and prove that its Gaussian curvature K and normal curvature K ^⊥ satisfy K + K ^⊥ = 4. We also show that the ellipse of curvature is a circle.

Keywords

Complex hyperquadric / Gaussian curvature / normal curvature / minimal immersion / ellipse of curvature

References

[1] Bolton J., Jensen G. R., Rigoli M., Woodward L. M. On conformal minimal immersions of S ₂ into ℂP_n. Math Ann, 1988, 279, 599-620

[2] Chern S. S. On the minimal immersions of the two-sphere in a space of constant curvature. Problems in Analysis, 1970, Princeton: Princeton Univ Press 27-40

[3] Chern S. S., Wolfson J. G. Minimal surfaces by moving frames. Amer J Math, 1983, 105, 59-83

[4] Eschenburg J. H., Guadalupe I. V., Tribuzy R. A. The fundamental equations of minimal surfaces in ℂP². Math Ann, 1985, 270, 571-598

[5] Houg C. S. Some totally real minimal surface in ℂP². Proc Amer Math Soc, 1973, 40, 240-244

[6] Jiao X. X., Peng J. G. Minimal 2-spheres in a complex projective space. Diff Geom Appl, 2007, 25, 506-517

[7] Jiao X. X., Wang J. Conformal minimal two-spheres in Qn. Sci China Math, 2011, 54 4 817-830

[8] Wolfson J. On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature. Trans Amer Math Soc, 1985, 290, 627-646

[9] Yang K. Complete and Compact Minimal Surface, 1989, Dordrecht: Kluwer Academic.