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Minimal surfaces in the complex hyperquadric $ Q_2$ II


Authors: Jun Wang and Xiaowei Xu
Journal: Proc. Amer. Math. Soc. 143 (2015), 2693-2703
MSC (2010): Primary 53C42, 53C55
DOI: https://doi.org/10.1090/S0002-9939-2015-12479-3
Published electronically: January 21, 2015
MathSciNet review: 3326047
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, minimal surfaces with parallel second fundamental form in $ Q_2$ are classified, which are uniquely determined up to a rigidity motion. It is also proved that minimal surfaces in $ Q_2$ with constant Gauss curvature and constant normal curvature are totally geodesic.


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  • [1] John Bolton, Gary R. Jensen, Marco Rigoli, and Lyndon M. Woodward, On conformal minimal immersions of $ S^2$ into $ {\bf C}{\rm P}^n$, Math. Ann. 279 (1988), no. 4, 599-620. MR 926423 (88m:53110), https://doi.org/10.1007/BF01458531
  • [2] Shigetoshi Bando and Yoshihiro Ohnita, Minimal $ 2$-spheres with constant curvature in $ {\rm P}_n({\bf C})$, J. Math. Soc. Japan 39 (1987), no. 3, 477-487. MR 900981 (88i:53097), https://doi.org/10.2969/jmsj/03930477
  • [3] Shiing Shen Chern and Jon Gordon Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), no. 1, 59-83. MR 692106 (84i:53056), https://doi.org/10.2307/2374381
  • [4] Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111-125. MR 0233294 (38 #1616)
  • [5] Dorel Fetcu, Surfaces with parallel mean curvature vector in complex space forms, J. Differential Geom. 91 (2012), no. 2, 215-232. MR 2971287
  • [6] Shinya Hirakawa, Constant Gaussian curvature surfaces with parallel mean curvature vector in two-dimensional complex space forms, Geom. Dedicata 118 (2006), 229-244. MR 2239458 (2007c:53073), https://doi.org/10.1007/s10711-005-9038-8
  • [7] Katsuei Kenmotsu and Kyûya Masuda, On minimal surfaces of constant curvature in two-dimensional complex space form, J. Reine Angew. Math. 523 (2000), 69-101. MR 1762956 (2001c:53086), https://doi.org/10.1515/crll.2000.053
  • [8] Katsuei Kenmotsu and Detang Zhou, The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms, Amer. J. Math. 122 (2000), no. 2, 295-317. MR 1749050 (2001a:53094)
  • [9] XiaoXiang Jiao and Jun Wang, Conformal minimal two-spheres in $ Q_n$, Sci. China Math. 54 (2011), no. 4, 817-830. MR 2786717 (2012b:53120), https://doi.org/10.1007/s11425-011-4179-8
  • [10] Jun Wang and Xiaoxiang Jiao, Conformal minimal two-spheres in $ Q_2$, Front. Math. China 6 (2011), no. 3, 535-544. MR 2802967 (2012e:53119), https://doi.org/10.1007/s11464-011-0137-6
  • [11] Xiaoxiang Jiao and Jun Wang, Minimal surfaces in a complex hyperquadric $ Q_2$, Manuscripta Math. 140 (2013), no. 3-4, 597-611. MR 3019141, https://doi.org/10.1007/s00229-012-0554-1
  • [12] Jun Wang and Xiaoxiang Jiao, Totally real minimal surfaces in the complex hyperquadrics, Differential Geom. Appl. 31 (2013), no. 4, 540-555. MR 3066034, https://doi.org/10.1016/j.difgeo.2013.05.007
  • [13] Takashi Ogata, Surfaces with parallel mean curvature vector in $ {\rm P}^2(\mathbf {C})$, Kodai Math. J. 18 (1995), no. 3, 397-407. MR 1362916 (96i:53061), https://doi.org/10.2996/kmj/1138043479
  • [14] Francisco Torralbo and Francisco Urbano, Surfaces with parallel mean curvature vector in $ {\mathbb{S}}^2\times {\mathbb{S}}^2$ and $ {\mathbb{H}}^2\times {\mathbb{H}}^2$, Trans. Amer. Math. Soc. 364 (2012), no. 2, 785-813. MR 2846353, https://doi.org/10.1090/S0002-9947-2011-05346-8
  • [15] Xu Zhong, Jun Wang, and XiaoXiang Jiao, Totally real conformal minimal tori in the hyperquadric $ Q_2$, Sci. China Math. 56 (2013), no. 10, 2015-2023. MR 3102623, https://doi.org/10.1007/s11425-013-4600-6
  • [16] Kichoon Yang, Complete and compact minimal surfaces, Mathematics and its Applications, vol. 54, Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1020302 (91h:53058)

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Additional Information

Jun Wang
Affiliation: School of Mathematics Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People’s Republic of China
Email: wangjun706@mails.ucas.ac.cn

Xiaowei Xu
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui province, People’s Republic of China; and Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, 230026, Anhui, People’s Republic of China
Email: xwxu09@ustc.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2015-12479-3
Keywords: Minimal surfaces, constant curvature, K\"ahler angle, totally geodesic
Received by editor(s): October 8, 2013
Received by editor(s) in revised form: January 27, 2014
Published electronically: January 21, 2015
Additional Notes: Xiaowei Xu served as corresponding author for this paper.
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2015 American Mathematical Society

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