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.
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Bolton J, Jensen G R, Rigoli M, Woodward L M. On conformal minimal immersions of S 2 into ℂPn. Math Ann, 1988, 279: 599–620
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
Chern S S, Wolfson J G. Minimal surfaces by moving frames. Amer J Math, 1983, 105: 59–83
Eschenburg J H, Guadalupe I V, Tribuzy R A. The fundamental equations of minimal surfaces in ℂP2. Math Ann, 1985, 270: 571–598
Houg C S. Some totally real minimal surface in ℂP2. Proc Amer Math Soc, 1973, 40: 240–244
Jiao X X, Peng J G. Minimal 2-spheres in a complex projective space. Diff Geom Appl, 2007, 25: 506–517
Jiao X X, Wang J. Conformal minimal two-spheres in Qn. Sci China Math, 2011, 54(4): 817–830
Wolfson J. On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature. Trans Amer Math Soc, 1985, 290: 627–646
Yang K. Complete and Compact Minimal Surface. Dordrecht: Kluwer Academic, 1989
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Wang, J., Jiao, X. Conformal minimal two-spheres in Q 2 . Front. Math. China 6, 535–544 (2011). https://doi.org/10.1007/s11464-011-0137-6
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DOI: https://doi.org/10.1007/s11464-011-0137-6