Frontiers of Mathematics in China >
Superminimal surfaces in hyperquadric Q2
Received date: 10 Jun 2020
Accepted date: 21 Sep 2020
Published date: 15 Oct 2020
Copyright
We study a superminimal surface M immersed into a hyperquadric Q2 in several cases classified by two global defined functions and , which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : . In case both and are not identically zero, it is proved that f is superminimal if and only if f is totally real or is also minimal, where is the standard inclusion map. In the rest case that or , 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
Jun WANG , Jie FEI . Superminimal surfaces in hyperquadric Q2[J]. Frontiers of Mathematics in China, 2020 , 15(5) : 1035 -1046 . DOI: 10.1007/s11464-020-0862-9
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 Qn. 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
|
/
〈 | 〉 |