RESEARCH ARTICLE

Minimal two-spheres with constant curvature in Pn

  • Shaoteng ZHANG ,
  • Xiaoxiang JIAO
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  • School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 07 Nov 2020

Accepted date: 17 Jan 2021

Published date: 15 Jun 2021

Copyright

2021 Higher Education Press

Abstract

We study conformal minimal two-spheres immersed into the quaternionic projective space Pn by using the twistor map. We present a method to construct new minimal two-spheres with constant curvature in Pn, based on the minimal property and horizontal condition of Veronese map in complex projective space. Then we construct some concrete examples of conformal minimal two-spheres in Pn with constant curvature 2/n, n = 4, 5, 6, respectively. Finally, we prove that there exist conformal minimal two-spheres with constant curvature 2/n in Pn (n≥7):

Cite this article

Shaoteng ZHANG , Xiaoxiang JIAO . Minimal two-spheres with constant curvature in Pn[J]. Frontiers of Mathematics in China, 2021 , 16(3) : 901 -923 . DOI: 10.1007/s11464-021-0902-0

1
Aithal A R. Harmonic maps from S2 to HP2. Osaka J Math, 1986, 23: 255–270

2
Bahy-El-Dien A, Wood J C. The explicit construction of all harmonic two-spheres in quaternionic projective spaces. Proc Lond Math Soc, 1991, 62: 202–224

DOI

3
Bolton J, Jensen G R, Rigoli M, Woodward L M. On conformal minimal immersion of S2 into CPn. Math Ann, 1988, 279: 599–620

DOI

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

DOI

5
Chen X D, Jiao X X. Conformal minimal surfaces immersed into HPn. Ann Mat Pura App, 2017, 196(6): 2063–2076

DOI

6
Fei J, He L. Classification of homogeneous minimal immersions from S2 to HPn. Ann Mat Pura Appl, 2017, 196(6): 2213–2237

DOI

7
Fei J, Peng C K, Xu X W. Minimal two-spheres with constant curvature in the quaternionic projective space. Sci China Math, 2020, 63(5): 993–1006

DOI

8
He L, Jiao X X. Classification of conformal minimal immersions of constant curvature from S2 to HP2. Math Ann, 2014, 359(3-4): 663–694

DOI

9
He L, Jiao X X. On conformal minimal immersions of S2 in HPn with parallel second fundamental form. Ann Mat Pura Appl, 2015, 194(5): 1301–1317

DOI

10
Jensen G R, Rigoli M, Yang K. Holomorphic curves in the complex quadric. Bull Aust Math Soc, 1987, 35(1): 125–148

DOI

11
Jiao X X, Cui H B. Construction of conformal minimal two-spheres in quaternionic projective spaces by twistor map. J Geom Anal, 2021, 31: 888–912

DOI

12
Jiao X X, Xu Y, Xin J L. Reducible minimal two-spheres of constant curvature in a quaternion projective space. Preprint

13
Yang K. Complete and Compact Minimal Surfaces. Dordrecht: Kluwer Academic Publishers, 1989

DOI

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