Abstract
We obtain a large number of new minimal 2-spheres of constant curvature in the quaternion projective space \({\mathbb{H}}\)Pn-1. Using the twistor fibration, we reduce the problem to constructing horizontal holomorphic spheres in the complex projective space \(\mathbb{C}P^{{2n - 1}}\). We prove that the set of such horizontal spheres is bijective to a closed disk consisting of certain anti-symmetric matrices modulo the action of \(U(1)\times SU(2)\). From this characterization, we deduce a lower bound on the dimension. Our method relies upon the singular decomposition analysis for the planes spanned by the spheres. Finally by checking the squared normal of the first \(\partial\)-return, we illustrate that the generic ones are not homogeneous, and thus not those that are classified.
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References
Bahy-El-Dien, A., Wood, J.C.: The explicit construction of all harmonic two-spheres in \(G_{2} \left( {\mathbb{R}^{n} } \right)\). J. Reine Angew. Math. 398, 36–66 (1989)
Bahy-El-Dien, A., Wood, J.C.: The explicit construction of all harmonic two-spheres in quaternionic projective spaces. Proc. Lond. Math. Soc. 62(1), 202–224 (1991)
Bolton, J., Jensen, G.R., Rigoli, M., Woodward, L.M.: On conformal minimal immersions of \(S^{2}\) into \(\mathbb{C}P^{n}\). Math. Ann. 279(4), 599–620 (1988)
Burstall, F.E., Wood, J.C.: The construction of harmonic maps into complex Grassmannians. J. Differ. Geom. 23, 255–297 (1986)
Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. Differ. Geom. 1, 111–125 (1967)
Chen, X.D., Jiao, X.X.: Conformal minimal surfaces immersed into \({\mathbb{H}}P^{n}\). Ann. Mat. Pura Appl. 196(6), 2063–2076 (2017)
Chern, S.S., Wolfson, J.G.: Harmonic maps of the two-sphere into a complex Grassmann manifold. II. Ann. Math. 2(125), 301–335 (1987)
Chi, Q.S., Xie, Z.X., Xu, Y.: Structure of minimal 2-spheres of constant curvature in the complex hyperquadric. arXiv:1903.11641
Fei, J., He, L.: Classification of homogeneous minimal immersions from \(S^{2}\) to \({\mathbb{H}}P^{n}\). Ann. Mat. Pura Appl. 196(6), 2213–2237 (2017)
Fei, J., Peng, C.K., Xu, X.W.: Minimal two-spheres with constant curvature in the quaternionic projective space. Sci, China Math (2019)
Gao, Z.J., Jiao, X.X.: The geometry of conformal minimal surface in \({\mathbb{H}}P^{3}\). J. Univ. Chin. Acad. Sci. 36(3), 299–310 (2019)
Harvey, F.R.: Spinors and calibrations. Academic Press Inc., Boston, MA etc (1990)
He, L., Jiao, X.X.: Classification of conformal minimal immersions of constant curvature from \(S^{2}\) to \(HP^{2}\). Math. Ann. 359(3–4), 663–694 (2014)
He, L., Jiao, X.X.: On conformal minimal immersions of constant curvature from \(S^{2}\) to \(HP^{n}\). Math. Z. 280(3–4), 851–871 (2015)
Horn, R.A., Johnson, C.R.: Matrix analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
Jiao, X.X., Cui, H.B.: Construction of Conformal Minimal Two-Spheres in Quaternionic Projective Spaces by Twistor Map. J. Geom. Anal. (2019)
Jiao, X.X., Xu, Y.: On non-\(\pm\)holomorphic conformal minimal two-spheres in a complex Grassmannian \(G(2,5)\) with constant curvature. Differ. Geom. Appl. 59, 154–183 (2018)
Kosmann-Schwarzbach, Y.: Groups and symmetries. From finite groups to Lie groups. Transl. from the French by Stephanie Frank Singer. New York, NY: Springer, (2010)
Ohnita, Y.: Homogeneous harmonic maps into complex projective spaces. Tokyo J. Math. 13(1), 87–116 (1990)
Reckziegel., H.: Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion. Global differential geometry and global analysis 1984, Proc. Conf., Berlin 1984, Lect. Notes Math. 1156, 264-279 (1985)
Acknowledgements
We are grateful to the anonymous referees for useful comments and suggestions. The second author would like to thank Prof. Q.S. Chi for encouragement and hospitality during his visit to WUSTL. This work was partially supported by NSFC No. 11871450. The second author was partially supported by China Post-Doctoral Grant BX20200012.
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Jiao, X., Xu, Y. & Xin, J. Minimal two-spheres of constant curvature in a quaternion projective space. Annali di Matematica 201, 1139–1155 (2022). https://doi.org/10.1007/s10231-021-01151-0
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DOI: https://doi.org/10.1007/s10231-021-01151-0
Keywords
- Constantly curved minimal 2-spheres
- Quaternion projective space
- Singular value decomposition
- Twistor fibration