Superminimal surfaces in hyperquadric Q 2
- PDF / 306,847 Bytes
- 12 Pages / 496.063 x 708.661 pts Page_size
- 64 Downloads / 177 Views
Superminimal surfaces in hyperquadric Q2 Jun WANG1 , Jie FEI2 1 School of Mathematics Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China 2 Department of Pure Mathematics, School of Science, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China
c Higher Education Press 2020
Abstract We study a superminimal surface M immersed into a hyperquadric Q2 in several cases classified by two global defined functions τX and τY , which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : M → Q2 . In case both τX and τY are not identically zero, it is proved that f is superminimal if and only if f is totally real or i◦f : M → CP3 is also minimal, where i : Q2 → CP3 is the standard inclusion map. In the rest case that τX ≡ 0 or τY ≡ 0, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q2 are completely described. Keywords Hyperquadric, superminimal surface, totally real, holomorphic MSC 53C42, 53C55 1
Introduction
Given a minimal immersion x : M → N from a connected and orientable surface M into a Riemannian manifold N, Chern and Wolfson [3] constructed a quartic differential form Q from the second fundamental form in a natural way. A minimal immersion x : M → N is called superminimal immersion if Q ≡ 0. The terminology ‘superminimal’ was introduced by Bryant [2], who determined all superminimal surfaces in the 4-sphere by using the twistor map of Penrose T : CP3 → S 4 . Almost at the same time, Chern and Wolfson derived Bryant’s result by moving frames. More precisely, when the ambient Riemannian manifold N is of constant curvature, Chern and Wolfson showed that the quartic form Q is holomorphic, relative to the underlying complex structure of M compatible with the orientation and induced metric on M. Especially, if M = S 2 , then the Riemann-Roch theorem implies that Q ≡ 0. Moreover, they also described all superminimal surfaces in the complex projective space CP2 with the Fubini-Study metric. The superminimal surfaces in general CPn had Received June 10, 2020; accepted September 21, 2020 Corresponding author: Jie FEI, E-mail: [email protected]
2
Jun WANG, Jie FEI
been studied by many authors (cf. [1,4,5,11]). We consider that the ambient manifold is the hyperquadric Q2 = {[Z = (z1 , z2 , z3 , z4 )T ] ∈ CP3 | z12 + z22 + z32 + z42 = 0}, which is isometric to the homogeneous space O(4)/(SO(2) × O(2)). Although Q2 is a complex algebraic submanifold of CP3 , it does not have constant holomorphic sectional curvature. As a consequence, the quartic form Q is not necessarily holomorphic, and so a minimal two-sphere in Q2 is not necessarily superminimal. Yang proposed that it is an interesting problem to describe the totality of superminimal surfaces or superminimal two-spheres in Q2 (cf. [14]). Wolfson [12] provided a description of harmonic maps from two-sphere into the complex hyperquadric Qn by applying harmonic sequences in the complex Grassmann manifold G(2, n, C). The geometry of minimal surfaces
Data Loading...
