The canonical lamination calibrated by a cohomology class

Abstract

Let $M$ be a closed oriented Riemannian manifold of dimension $2\le d\le 7$, 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 Teichmüller space.

27 pages; this is the journal version; comments welcome