Stability of Diophantine Minimal Foliations in Arbitrary Codimension

Abstract

We generalize Moser's stability theorem for minimal hypersurface foliations to arbitrary codimension. Let ${\mathbb{T}}^N={\mathbb{R}}^N/{\mathbb{Z}}^N$ be the standard torus, and let $P\subset {\mathbb{R}}^N$ be an $m$-plane satisfying a Diophantine condition relative to ${\mathbb{Z}}^N$. We prove that, under every sufficiently small smooth ${\mathbb{Z}}^N$-periodic perturbation of the Euclidean metric, the affine foliation of ${\mathbb{T}}^N$ induced by $P$ persists as a smoothly conjugate foliation by minimal $m$-dimensional submanifolds. The normalized conjugating displacement depends smoothly and tamely on the metric and is locally unique. Moreover, the perturbed foliation admits a common smooth calibration, hence the lifted leaves are area-minimizing. The proof uses Hamilton's formulation of the Nash-Moser implicit function theorem.

arXiv comment38 pages