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Calibrations and laminations. (English) Zbl 1381.53091

Summary: A calibration of degree \(k\in\mathbb{N}\) on a Riemannian manifold \(M\) is a closed differential \(k\)-form \(\theta\) such that the integral of \(\theta\) over every \(k\)-dimensional, oriented submanifold \(N\) is smaller or equal to the Riemannian volume of \(N\). A calibration \(\theta\) is said to calibrate \(N\) if \(\theta\) restricts to the oriented volume form of \(N\). We investigate conditions on a calibration \(\theta\) that ensure the existence of submanifolds calibrated by \(\theta\). The cases \(k = 1\) and \(k > 1\) turn out to be essentially different. Our main result says that, on a compact manifold \(M\), a calibration \(\theta\) calibrates a lamination if \(\theta\) is simple, of class \(C^1\), and if \(\theta\) has minimal comass norm in its cohomology class.

MSC:

53C38 Calibrations and calibrated geometries
53C12 Foliations (differential geometric aspects)
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