Abstract
This paper is the continuation of the previous one Jiao and Cui (Area-Minimizing Cones Over Grassmannian Manifolds. J. Geom. Anal. 32, 224 (2022). https://doi.org/10.1007/s12220-022-00963-7), where we re-proved the area-minimization of cones over Grassmannians of n-planes \(G(n,m;{\mathbb {F}})({\mathbb {F}}={\mathbb {R}},{\mathbb {C}},{\mathbb {H}})\), Cayley plane \({\mathbb {O}}P^2\) from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians \({\widetilde{G}}(n,m;{\mathbb {R}})\) by using Lawlor’s Curvature Criterion Lawlor (Mem Amer Math Soc 91(446), 1991). Here, we make a further step on showing that the cones, of dimension no less than \({\mathbf {8}}\), over minimal products of \(G(n,m;{\mathbb {F}})\) are area-minimizing. Moreover, those cones are very similar to the classical cones over products of spheres, and for the critical situation—the cones of dimension \({\mathbf {7}}\) Lawlor (Mem Amer Math Soc 91(446), 1991), we gain more area-minimizing cones by carefully computing the Jacobian \(inf_{v}det(I-tH^{v}_{ij})\). Certain minimizing cones among them had been found from the perspective of R-spaces Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), or isoparametric theory Tang and Zhang (J Differ Geom 115(2):367–393, 2020) recently, and the generic ones in our results are completely new. We also prove that the cones over minimal product of general \({\widetilde{G}}(n,m;{\mathbb {R}})\) are area-minimizing, it can be seen as generalized results for some \({\widetilde{G}}(2,m;{\mathbb {R}})\) shown in Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), Tang and Zhang (J Differ Geom 115(2):367–393, 2020).
Similar content being viewed by others
References
Berndt, J., Console, S., Olmos, C.E.: Submanifolds and holonomy, vol. 21. Chapman and Hall/CRC., 2nd ed. edition (2016)
Cheng, B.N.: Area-minimizing Cone-type Surfaces and Coflat Calibrations. Indiana Univ. Math. J. 37(3), 505–535 (1988)
Chen, B.-Y.: Total mean curvature and submanifolds of finite type, vol. 27. World Scientific (1984)
Choe, J., Hoppe, J.: Some minimal submanifolds generalizing the Clifford torus. Math. Nachr. 291, 2536–2542 (2018)
Harvey, F.R.: Spinors and calibrations. Elsevier, (1990)
Hirohashi, D., Kanno, T., Tasaki, H.: Area-minimizing of the cone over symmetric R-spaces. Tsukuba J. Math. 24(1), 171–188 (2000)
Jiao, X., Cui, H.: Area-Minimizing Cones Over Grassmannian Manifolds. J. Geom. Anal. 32, 224 (2022). https://doi.org/10.1007/s12220-022-00963-7
Kanno, T.: Area-minimizing cones over the canonical embedding of symmetric R-spaces. Indiana Univ. Math. J. 51(1), 89–125 (2002)
Kerckhove, M.: Isolated orbits of the adjoint action and area-minimizing cones. Proc. Amer. Math. Soc. 121(2), 497–503 (1994)
Lawlor, G.R.: A sufficient criterion for a cone to be area-minimizing. Mem. Amer. Math. Soc., 91(446), (1991)
Lawlor, G.R., Murdoch, T.A.: A note about the Veronese cone. Illinois J. Math. 39(2), 271–277 (1995)
Morgan, F.: Calibrations modulo v. Adv. Math. 64(1), 32–50 (1987)
Murdoch, T.A.: Twisted calibrations. Trans. Amer. Math. Soc. 328(1), 239–257 (1991)
Ohno, S., Sakai, T.: Area-minimizing cones over minimal embeddings of R-spaces. Josai Math. Monogr. 13, 69–91 (2021)
Tang, Z., Zhang, Y.: Minimizing cones associated with isoparametric foliations. J. Differ. Geom. 115(2), 367–393 (2020)
Xiaowei, X., Yang, L., Zhang, Y.: New area-minimizing Lawson-Osserman cones. Adv. Math. 330, 739–762 (2018)
Acknowledgements
All authors are indebted to Professor Yongsheng Zhang for some valuable suggestions, and we would like to thank the referees for countless helpful advice on both the structure of the paper and the writing of some proofs, which extremely improve the readability of the paper. This work was supported in part by NSFC(Grant Number 11871450), and the project of Stable Support for Youth Team in Basic Research Field, CAS(YSBR-001).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Jost.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jiao, X., Cui, H. & Xin, J. Area-minimizing cones over products of Grassmannian manifolds. Calc. Var. 61, 205 (2022). https://doi.org/10.1007/s00526-022-02309-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-022-02309-1