ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: not given Gu, Weiqing, Harvey Mudd College, Claremont, USA The stable 4-dimensional geometry of the real Grassmann manifolds In their fundamental paper on Calibrated geometries in 1982, Reese Harvey and Blaine Lawson, discussing future global applications, posed the problem of determining the subvarietis of the real Grassmann manifolds G_{k}R^{n} of oriented k-planes through the origin in R^{n}, which can be shown to be volumeminimizing in their homology classes by using as calibrations the invariant forms representing the universal Pontryagin classes. We carray out this program here for the first Pontryagin form, using it as a calibration to determine volume-minimizing 4-dimensional cycles in all real Grassmann manifolds, and learning that their 4-dimensional geometry stabilizes immediately after the Grassmannian G_{4}R^{8}.