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 | References | Similar Articles | Additional Information

One way to understand the geometry of the real Grassmann manifold parameterizing oriented -dimensional subspaces of 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 -planes calibrated by the first Pontryagin form on for all , and identified a family of mutually congruent round -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 , an analysis of which yields a uniqueness result; namely, that any connected submanifold of calibrated by is contained in one of these -spheres. A similar result holds for -calibrated submanifolds of the quotient Grassmannian of non-oriented -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:
https://doi.org/10.1090/S0002-9947-01-02783-0

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