Superminimal surfaces in hyperquadric Q2

Abstract

We study a superminimal surface M immersed into a hyperquadric Q₂ in several cases classified by two global defined functions {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y, which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : M\to {{\displaystyle Q}}_{\mathrm{2}}. In case both {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y are not identically zero, it is proved that f is superminimal if and only if f is totally real or i\circ f:M\to {ℂ\mathrm{P}}^{\mathrm{3}} is also minimal, where i:{{\displaystyle Q}}_{\mathrm{2}}\to {ℂ\mathrm{P}}^{\mathrm{3}} is the standard inclusion map. In the rest case that {{\displaystyle \tau }}_X\equiv \mathrm{0} or {{\displaystyle \tau }}_Y\equiv \mathrm{0}, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q₂ are completely described.

Key words： Hyperquadric; superminimal surface; totally real; holomorphic

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 Bryant R L. Conformal and minimal immersions of compact surfaces into the 4-sphere. J Differential Geom, 1982, 17: 455–473

3 Chern S S, Wolfson J G. Minimal surfaces by moving frames. Amer J Math, 1983, 105: 59–83

4 Din A M, Zakrzewski W J. General classical solutions in the ℂPn model. Nuclear Phys B, 1980, 174: 397–406

5 Eells J, Wood J C. Harmonic maps from surfaces to complex projective spaces. Adv Math, 1983, 49: 217–263

6 Fei J, Wang J. Local rigidity of minimal surfaces in a hyperquadric Q₂. J Geom Phys, 2018, 133: 17–25

7 Jiao X X, Wang J. Conformal minimal two-spheres in Q_n. Sci China Math, 2011, 54(4): 817–830

8 Jiao X X, Wang J. Minimal surfaces in a complex hyperquadric Q₂. Manuscripta Math, 2013, 140: 597–611

9 Peng C K, Wang J, Xu X W. Minimal two-spheres with constant curvature in the complex hyperquadric. J Math Pures Appl, 2016, 106: 453–476

10 Wang J, Xu X W. Minimal surfaces in the complex hyperquadric Q₂ II. Proc Amer Math Soc, 2015, 143: 2693–2703

11 Wolfson J G. On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature. Trans Amer Math Soc, 1985, 290(2): 597–611

12 Wolfson J G. Harmonic maps of the two-sphere into the complex hyperquadric. J Differential Geom, 1986, 24: 141–152

13 Yang K. Frenet formulae for holomorphic curves in the two quadric. Bull Aust Math Soc, 1986, 33: 195–206

14 Yang K. Complete and Compact Minimal Surfaces. Dordrecht: Kluwer Academic, 1989

15 Zhong X, Wang J, Jiao X X. Totally real conformal minimal tori in the hyperquadric Q₂. Sci China Math, 2013, 56: 2015–2023