The canonical lamination calibrated by a cohomology class

Abstract

Let $M$ be a closed oriented Riemannian manifold of dimension $2 \leq d \leq 7$, and let $\rho \in H^{d - 1}(M, \mathbb R)$ have unit norm. We construct a lamination $\lambda_\rho$ whose leaves are exactly the minimal hypersurfaces which are calibrated by every calibration in $\rho$. The geometry of $\lambda_\rho$ is closely related to the the geometry of the unit ball of the stable norm on $H_{d - 1}(M, \mathbb R)$, and so we deduce several results constraining the geometry of the stable norm ball in terms of the topology of $M$. These results establish a close analogy between the stable norm on $H_{d - 1}(M, \mathbb R)$ and the earthquake norm on the tangent space to Teichm\"uller space.