Conformal minimal two-spheres in Q₂

Abstract

In this paper, we consider a conformal minimal immersion f from S² into a hyperquadric Q₂, 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 S² into ℂPn. Math Ann, 1988, 279: 599-620

[2] Chern S S. On the minimal immersions of the two-sphere in a space of constant curvature. In: Problems in Analysis. Princeton: Princeton Univ Press, 1970, 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 ℂPn. Math Ann, 1985, 270: 571-598

[5] Houg C S. Some totally real minimal surface in ℂPn. 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 Q_n. 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. Dordrecht: Kluwer Academic, 1989

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg