Totally real flat minimal surfaces in the hyperquadric
Ling He
Center for Applied Mathematics, Tianjin University, Tianjin, China
Search for more papers by this authorXiaoxiang Jiao
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
Search for more papers by this authorCorresponding Author
Mingyan Li
School of Mathematical Sciences, Ocean University of China, Qingdao, China
Correspondence
Mingyan Li, School of Mathematical Sciences, Ocean University of China, Qingdao 266000, China.
Email: [email protected]
Search for more papers by this authorLing He
Center for Applied Mathematics, Tianjin University, Tianjin, China
Search for more papers by this authorXiaoxiang Jiao
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
Search for more papers by this authorCorresponding Author
Mingyan Li
School of Mathematical Sciences, Ocean University of China, Qingdao, China
Correspondence
Mingyan Li, School of Mathematical Sciences, Ocean University of China, Qingdao 266000, China.
Email: [email protected]
Search for more papers by this authorAbstract
In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric , and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both and , we determine them for , and give a classification theorem when they are Clifford solutions.
REFERENCES
- 1A. Bahy-El-Dien and J. C. Wood, The explicit construction of all harmonic two-spheres in , J. Reine Angew. Math. 398 (1989), 36–66.
- 2J. Bolton, G. R. Jensen, M. Rigoli, and L. M. Woodward, On conformal minimal immersions of S2 into , Math. Ann. 279 (1988), 599–620.
- 3J. Bolton and L. M. Woodward, Minimal surfaces in with constant curvature and Kähler angle, Proc. Cambridge Philos. Soc. 112 (1992), 287–296.
- 4R. L. Bryant, Minimal surfaces of constant curvature in , Trans. Amer. Math. Soc. 290 (1985), 259–271.
- 5F. E. Burstall and J. C. Wood, The construction of harmonic maps into complex Grassmannians, J. Diff. Geom. 23 (1986), 255–297.
- 6I. Castro and F. Urbano, Minimal Lagrangian surfaces in , Comm. Anal. Geom. 15 (2007), 217–248.
- 7S. S. Chern and J. G. Wolfson, Minimal surfaces by moving frames, Am. J. Math. 105 (1983), 59–83.
- 8S. S. Chern and J. G. Wolfson, Harmonic maps of the two-spheres in a complex Grassmann manifold , Ann. Math. 125 (1987), 301–335.
- 9Q. S. Chi, Z. X. Xie, and Y. Xu, Structure of minimal 2-spheres of constant curvature in the complex hyperquadric, Adv. Math. 391 (2021), 34pp.
- 10P. J. Davis, Circulant matrices, American Mathematical Society, 2nd ed., 2012.
- 11J. Fei and J. Wang, Local rigidity of minimal surfaces in a hyperquadric Q2, J. Geom. Phys. 133 (2018), 17–25.
- 12G. R. Jensen and R. J. Liao, Families of flat minimal tori in , J. Diff. Geom. 42 (1995), no. 1, 113–132.
- 13G. R. Jensen, M. Rigoli, and K. Yang, Holomorphic curves in the complex quadric, Bull. Austral. Math. Soc. 35 (1987), no. 1, 125–148.
- 14X. X. Jiao, On minimal two-spheres immersed in complex Grassmann manifolds with parallel second fundamental form, Monatsh. Math. 168 (2012), 381–401.
- 15X. X. Jiao and M. Y. Li, Classification of conformal minimal immersions of constant curvature from S2 to , Ann. Mat. 196 (2017), 1001–1023.
- 16X. X. Jiao and M. Y. Li, On conformal minimal immersions of two-spheres in a complex hyperquadric with parallel second fundamental form, J. Geom. Anal. 26 (2016), 185–205.
- 17K. Kenmotsu, On minimal immersions of into , J. Math. Soc. Japan 37 (1985), 665–682.
- 18H. Z. Li, H. Ma, J. V. Veken, L. Vrancken, and X. F. Wang, Minimal Lagrangian submanifolds of the complex hyperquadric, Sci. China Math. 63 (2020), 1441–1462.
- 19M. Y. Li, X. X. Jiao, and L. He, Classification of conformal minimal immersions of constant curvature from S2 to Q3, J. Math. Soc. Japan 68 (2016), no. 2, 863–883.
- 20R. Liao, Cyclic properties of the harmonic sequence of surfaces in , Math. Ann. 296 (1993), 363–384.
- 21R. Pacheco and J. C. Wood, Diagrams and harmonic maps, revisited, Ann. Mat. Pura Appl. 202 (2023), 1051–1085.
- 22C. K. Peng, J. Wang, and X. W. Xu, Minimal two-spheres with constant curvature in the complex hyperquadric, J. Math. Pures Appl. (9) 106 (2016), 453–476.
- 23K. Uhlenbeck, Harmonic maps into Lie groups:classical solutions of the chiral model, J. Diff. Geom. 30 (1989), 1–50.
- 24J. G. Wolfson, Harmonic sequences and harmonic maps of surfaces into complex Grassmann manifolds, J. Diff. Geom. 27 (1988), 161–178.
- 25J. G. Wolfson, Harmonic maps of the two-sphere into the complex hyperquadric, J. Diff. Geom. 24 (1986), 141–152.
- 26J. C. Wood, The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian, J. Reine Angew. Math. 386 (1988), 1–31.
