Area-minimizing cones over Grassmannian manifolds. (English) Zbl 1519.53046
An important class of area-minimizing surfaces is represented by the area-minimizing cones. Their truncated part inside the unit ball minimizes the area among all integral currents with the same boundary. The existence of an isolated singularity at the origin is a special property of the area-minimizing cones, furnishing examples of solutions of noninterior regularity to Plateau’s problem which refers to the existence of an area-minimizing surface, bounded by a given Jordan curve in the Euclidean space.
Investigating the necessary and sufficient conditions for a cone to be area-minimizing, G. R. Lawlor provided in [A sufficient criterion for a cone to be area-minimizing. Providence, RI: American Mathematical Society (AMS) (1991; Zbl 0745.49029)] the Curvature Criterion (a general method for proving that a cone is area-minimizing). This method was used to prove the area-minimization of cones from the point of view of isolated orbits of adjoint actions or canonical embeddings of symmetric spaces (see [T. Kanno, Indiana Univ. Math. J. 51, No. 1, 89–125 (2002; Zbl 1035.53072); M. Kerckhove, Proc. Am. Math. Soc. 121, No. 2, 497–503 (1994; Zbl 0804.49031); S. Ohno and T. Sakai, “Area-minimizing cones over minimal embeddings of R-spaces”, Preprint, arXiv:1507.02006]). These results are re-proved in this paper: the authors describe first the standard minimal embedding map for the Grassmannians of \(n\)-planes in \(\mathbb{F}^m\) and the Cayley plane into Euclidean spheres by seeing \(G(n, m; \mathbb{F})\) \((\)where \(\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H})\) and \(\mathbb{O}P^2\) as Hermitian orthogonal projectors uniformly. It is shown that the Grassmannians \(G(n, m; \mathbb{F})\) and \( G(m - n, m; \mathbb{F}),\) where \(\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}\), can be embedded simultaneously into the same one Euclidean sphere as a pair of opposite minimal submanifolds and their images are two opposite area-minimizing cones. The area-minimization of cones over projective spaces is proved and then, seeing the Cayley plane \(\mathbb{O}P^2\) as the set of Hermitian orthogonal projection operators, it is shown that this cone is area-minimizing.
The last section of the paper deals with cones over oriented real Grassmannians \(\widetilde{G}(n, m; \mathbb{R})\), seen as unit simple vectors in the exterior vector spaces. The authors show that the cone over \(\widetilde{G}(2, 4; \mathbb{R})\) is unstable and the cones over the Plücker embeddings of all the other oriented real Grassmannians are area-minimizing.
Investigating the necessary and sufficient conditions for a cone to be area-minimizing, G. R. Lawlor provided in [A sufficient criterion for a cone to be area-minimizing. Providence, RI: American Mathematical Society (AMS) (1991; Zbl 0745.49029)] the Curvature Criterion (a general method for proving that a cone is area-minimizing). This method was used to prove the area-minimization of cones from the point of view of isolated orbits of adjoint actions or canonical embeddings of symmetric spaces (see [T. Kanno, Indiana Univ. Math. J. 51, No. 1, 89–125 (2002; Zbl 1035.53072); M. Kerckhove, Proc. Am. Math. Soc. 121, No. 2, 497–503 (1994; Zbl 0804.49031); S. Ohno and T. Sakai, “Area-minimizing cones over minimal embeddings of R-spaces”, Preprint, arXiv:1507.02006]). These results are re-proved in this paper: the authors describe first the standard minimal embedding map for the Grassmannians of \(n\)-planes in \(\mathbb{F}^m\) and the Cayley plane into Euclidean spheres by seeing \(G(n, m; \mathbb{F})\) \((\)where \(\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H})\) and \(\mathbb{O}P^2\) as Hermitian orthogonal projectors uniformly. It is shown that the Grassmannians \(G(n, m; \mathbb{F})\) and \( G(m - n, m; \mathbb{F}),\) where \(\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}\), can be embedded simultaneously into the same one Euclidean sphere as a pair of opposite minimal submanifolds and their images are two opposite area-minimizing cones. The area-minimization of cones over projective spaces is proved and then, seeing the Cayley plane \(\mathbb{O}P^2\) as the set of Hermitian orthogonal projection operators, it is shown that this cone is area-minimizing.
The last section of the paper deals with cones over oriented real Grassmannians \(\widetilde{G}(n, m; \mathbb{R})\), seen as unit simple vectors in the exterior vector spaces. The authors show that the cone over \(\widetilde{G}(2, 4; \mathbb{R})\) is unstable and the cones over the Plücker embeddings of all the other oriented real Grassmannians are area-minimizing.
Reviewer: Simona Druta-Romaniuc (Iaşi)
MSC:
| 53C35 | Differential geometry of symmetric spaces |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 49Q05 | Minimal surfaces and optimization |
References:
| [1] | Jurgen Berndt, Sergio Console, and Carlos Enrique Olmos. Submanifolds and holonomy, volume 21. Chapman and Hall/CRC., 2nd ed. edition, (2016) · Zbl 1448.53001 |
| [2] | Jeff, Cheeger., David G, Ebin.: Comparison theorems in Riemannian geometry.Revised reprint of the 1975 original., volume 365. AMS Chelsea Pub., (2008) · Zbl 1142.53003 |
| [3] | Bang-Yen, Chen.: Total mean curvature and submanifolds of finite type, volume 27. World Scientific, (1984) · Zbl 0537.53049 |
| [4] | Chen, Wei-Huan, Geometry of grassmannian manifolds as submanifolds(in chinese), Acta Math. Sin., 31, 1, 46-53 (1988) · Zbl 0672.53026 |
| [5] | Cheng, Benny N., Area-minimizing Cone-type Surfaces and Coflat Calibrations, Indiana Univ. Math. J., 37, 3, 505-535 (1988) · Zbl 0644.53006 · doi:10.1512/iumj.1988.37.37026 |
| [6] | Shiing-shen Chern and Jon Gordon Wolfson, Minimal surfaces by moving frames, Amer. J. Math., 105, 1, 59-83 (1983) · Zbl 0521.53050 · doi:10.2307/2374381 |
| [7] | Dadok, Jiri; Harvey, Reese, The pontryagin 4-form, Proc. Amer. Math. Soc., 127, 11, 3175-3180 (1999) · Zbl 0943.53035 · doi:10.1090/S0002-9939-99-05443-X |
| [8] | Federer, Herbert.: Geometric measure theory. Springer-Verlag, (1969) · Zbl 0176.00801 |
| [9] | Grossman, Daniel; Weiqing, Gu, Uniqueness of volume-minimizing submanifolds calibrated by the first pontryagin form, Trans. Amer. Math. Soc., 353, 11, 4319-4332 (2001) · Zbl 1031.53080 · doi:10.1090/S0002-9947-01-02783-0 |
| [10] | Gluck, Herman; Mackenzie, Dana; Morgan, Frank, Volume-minimizing cycles in grassmann manifolds, Duke Math. J., 79, 2, 335-404 (1995) · Zbl 0837.53035 · doi:10.1215/S0012-7094-95-07909-5 |
| [11] | Gluck, Herman; Morgan, Frank; Ziller, Wolfgang, Calibrated geometries in grassmann manifolds, Comment. Math. Helv., 64, 1, 256-268 (1989) · Zbl 0681.53039 · doi:10.1007/BF02564674 |
| [12] | Griffiths, Phillip, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J., 41, 4, 775-814 (1974) · Zbl 0294.53034 · doi:10.1215/S0012-7094-74-04180-5 |
| [13] | Weiqing, Gu, The stable 4-dimensional geometry of the real grassmann manifolds, Duke Math. J., 93, 1, 155-178 (1998) · Zbl 0943.53036 |
| [14] | Harvey, F Reese.: Spinors and calibrations. Elsevier, (1990) · Zbl 0694.53002 |
| [15] | Hirohashi, Daigo; Kanno, Takahiro; Tasaki, Hiroyuki, Area-minimizing of the cone over symmetric R-spaces, Tsukuba J. Math., 24, 1, 171-188 (2000) · Zbl 0991.53033 · doi:10.21099/tkbjm/1496164053 |
| [16] | Reese, Harvey; H Blaine, Lawson, Calibrated geometries, Acta Math, 148, 1, 47-157 (1982) · Zbl 0584.53021 |
| [17] | Jiao, Xiaoxiang., Cui, Hongbin., Xin, Jialin.: Area-minimizing cones over products of grassmannian manifolds. arXiv preprint arXiv: 2102.05337, 2021 |
| [18] | Takahiro, Kanno.: Area-minimizing cones over the canonical embedding of symmetric R-spaces. Indiana Univ. Math. J., pages 89-125, (2002) · Zbl 1035.53072 |
| [19] | Kerckhove, Michael, Isolated orbits of the adjoint action and area-minimizing cones, Proc. Amer. Math. Soc., 121, 2, 497-503 (1994) · Zbl 0804.49031 · doi:10.1090/S0002-9939-1994-1196166-6 |
| [20] | Kobayashi, Shoshichi, Isometric imbeddings of compact symmetric spaces, Tohoku Math. J. (2), 20, 1, 21-25 (1968) · Zbl 0175.48301 · doi:10.2748/tmj/1178243214 |
| [21] | Blaine Lawson, Jr. H.: Complete minimal surfaces in S3. Ann. of Math., pages 335-374, (1970) · Zbl 0205.52001 |
| [22] | H Blaine Lawson. The equivariant Plateau problem and interior regularity. Trans. Amer. Math. Soc., 173:231-249, 1972 · Zbl 0279.49043 |
| [23] | Gary Reid, Lawlor.: A sufficient criterion for a cone to be area-minimizing. Mem. Amer. Math. Soc., 91(446), (1991) · Zbl 0745.49029 |
| [24] | Lawlor, Gary R.; Murdoch, Timothy A., A note about the Veronese cone, Illinois J. Math, 39, 2, 271-277 (1995) · Zbl 0840.49019 · doi:10.1215/ijm/1255986548 |
| [25] | Lawson, H. Blaine; Robert, Osserman, Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system, Acta Math., 139, 1, 1-17 (1977) · Zbl 0376.49016 · doi:10.1007/BF02392232 |
| [26] | Morgan, Frank, The exterior algebra \(\lambda\) krn and area minimization, Linear Algebra Appl., 66, 1-28 (1985) · Zbl 0585.49029 · doi:10.1016/0024-3795(85)90123-5 |
| [27] | Morgan, Frank, Calibrations modulo v, Adv. Math., 64, 1, 32-50 (1987) · Zbl 0618.49019 · doi:10.1016/0001-8708(87)90003-X |
| [28] | Morgan, Frank.: Geometric measure theory: a beginner’s guide. Fifth edition. Academic press, (2016) · Zbl 1338.49089 |
| [29] | Murdoch, Timothy A., Twisted calibrations, Trans. Amer. Math. Soc, 328, 1, 239-257 (1991) · Zbl 0748.53034 · doi:10.1090/S0002-9947-1991-1069738-9 |
| [30] | Shinji Ohno and Takashi Sakai. Area-minimizing cones over minimal embeddings of R-spaces. arXiv preprint arXiv: 1507.02006, (2015) |
| [31] | Vincent Pavan. Exterior Algebras: Elementary Tribute to Grassmann’s Ideas. Elsevier, (2017) · Zbl 1377.15002 |
| [32] | Leiba Rodman. Topics in quaternion linear algebra, volume 45. Princeton University Press, 2014 · Zbl 1304.15004 |
| [33] | Sakamoto, Kunio, Planar geodesic immersions, Tohoku Math. J.(2), 29, 1, 25-56 (1977) · Zbl 0357.53035 · doi:10.2748/tmj/1178240693 |
| [34] | Tang, Zizhou; Zhang, Yongsheng, Minimizing cones associated with isoparametric foliations, J. Differential Geom., 115, 2, 367-393 (2020) · Zbl 1441.49035 · doi:10.4310/jdg/1589853628 |
| [35] | Uhlenbeck, Karen, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom., 30, 1, 1-50 (1989) · Zbl 0677.58020 · doi:10.4310/jdg/1214443286 |
| [36] | Xiaowei, Xu; Yang, Ling; Zhang, Yongsheng, New area-minimizing Lawson-Osserman cones, Adv. Math., 330, 739-762 (2018) · Zbl 1393.53051 · doi:10.1016/j.aim.2018.03.021 |
| [37] | Zhang, Fuzhen, Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57 (1997) · Zbl 0873.15008 · doi:10.1016/0024-3795(95)00543-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
