Uniqueness of Volume-Minimizing Submanifolds Calibrated by the First Pontryagin Form

Authors

Daniel A. Grossman, Princeton University
Weiqing Gu, Harvey Mudd CollegeFollow

Document Type

Article

Department

Mathematics (HMC)

Publication Date

6-2001

Abstract

One way to understand the geometry of the real Grassmann manifold G_k(R^k+n) parameterizing oriented k-dimensional subspaces of 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 p₁ on G_k(R^k+n) for all k,n ≥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, R), an analysis of which yields a uniqueness result; namely that any connected submanifold of G_k(R^k+n) calibrated by p₁ is contained in one of these 4-spheres. A similar result holds for p₁-calibrated submanifolds of the quotient Grassmannian G_k^♮(R^k+n) of non-oriented k-planes.

Comments

First published in Transactions of the American Mathematical Society 353 (2001), 4319-4332, published by the American Mathematical Society.

Rights Information

© 2001 American Mathematical Society

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

DOI

10.1090/S0002-9947-01-02783-0

Recommended Citation

Grossman, Daniel A.; Gu, Weiqing. "Uniqueness of volume-minimizing submanifolds calibrated by the first Pontryagin form." Trans. Amer. Math. Soc. 353 (2001), no. 11, 4319–4332.