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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Uniqueness of volume-minimizing submanifolds calibrated by the first Pontryagin form


Authors: Daniel A. Grossman and Weiqing Gu
Journal: Trans. Amer. Math. Soc. 353 (2001), 4319-4332
MSC (2000): Primary 53C38; Secondary 58A17, 53C40
Published electronically: June 14, 2001
MathSciNet review: 1851172
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Abstract:

One way to understand the geometry of the real Grassmann manifold $G_k(\mathbf{R}^{k+n})$ parameterizing oriented $k$-dimensional subspaces of $\mathbf{R}^{k+n}$ is to understand the volume-minimizing subvarieties in each homology class. Some of these subvarieties can be determined by using a calibration. In previous work, one of the authors calculated the set of $4$-planes calibrated by the first Pontryagin form $p_1$ on $G_k(\mathbf{R}^{k+n})$for all $k,n\geq 4$, and identified a family of mutually congruent round $4$-spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of calibrated planes a Pfaffian system on the symmetry group $SO(k+n,\mathbf R)$, an analysis of which yields a uniqueness result; namely, that any connected submanifold of $G_k(\mathbf{R}^{k+n})$ calibrated by $p_1$ is contained in one of these $4$-spheres. A similar result holds for $p_1$-calibrated submanifolds of the quotient Grassmannian $G_k^\natural(\mathbf{R}^{k+n})$ of non-oriented $k$-planes.


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Additional Information

Daniel A. Grossman
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Address at time of publication: Deparment of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637
Email: dan@math.uchicago.edu

Weiqing Gu
Affiliation: Department of Mathematics, Harvey Mudd College, Claremont, California 91711
Email: gu@math.hmc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02783-0
PII: S 0002-9947(01)02783-0
Keywords: Calibrated geometry, Pontryagin form, Pfaffian systems
Received by editor(s): April 1, 2000
Received by editor(s) in revised form: September 23, 2000
Published electronically: June 14, 2001
Additional Notes: The first author’s research was supported by a fellowship from the Alfred P. Sloan foundation
Article copyright: © Copyright 2001 American Mathematical Society