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

 

Calibrations on $ {\bf R}\sp 8$


Authors: J. Dadok, R. Harvey and F. Morgan
Journal: Trans. Amer. Math. Soc. 307 (1988), 1-40
MSC: Primary 53C42; Secondary 15A75, 49F10, 58A10, 58E15
MathSciNet review: 936802
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Abstract: Calibrations are a powerful tool for constructing minimal surfaces. In this paper we are concerned with $ 4$-manifolds $ M \subset {{\mathbf{R}}^8}$. If a differential form $ \varphi \in { \bigwedge ^4}{{\mathbf{R}}^8}$ calibrates all tangent planes of $ M$ then $ M$ is area minimizing. For $ \varphi $ in one of several large subspaces of $ { \bigwedge ^4}{{\mathbf{R}}^8}$ we compute its comass, that is the maximal value of $ \varphi $ on the Grassmannian of oriented $ 4$-planes. We then determine the set $ G(\varphi ) \subset G(4,\,{{\mathbf{R}}^8})$ on which this maximum is attained. These are the planes calibrated by $ \varphi $, the possible tangent planes to a manifold calibrated by $ \varphi $. The families of calibrations obtained include the well-known examples: special Lagrangian, Kähler, and Cayley.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0936802-1
PII: S 0002-9947(1988)0936802-1
Article copyright: © Copyright 1988 American Mathematical Society