Calibrations on 𝑅⁸

Dadok, J.; Harvey, R.; Morgan, F.

doi:10.1090/S0002-9947-1988-0936802-1

Calibrations on $\textbf {R}^ 8$
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by J. Dadok, R. Harvey and F. Morgan

Trans. Amer. Math. Soc. 307 (1988), 1-40

DOI: https://doi.org/10.1090/S0002-9947-1988-0936802-1

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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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Least-volume representatives of homology classes in