Jtdylktuy

Let $M$ be a closed connected surface and $mathcal{F}$ a minimal (every leaf is dense) measured foliation (as, for example, in Thurston's work on surfaces) on $M$. Let $tilde{M}$ be the universal cover of $M$ and $tilde{mathcal{F}}$ the pullback of $mathcal{F}$ to $tilde{M}$.

I saw a claim saying "every regular leaf (a leaf in $tilde{mathcal{F}}$ that does not contain a separatrix or the union of a singularity and all separatrices ending in that singularity) is a closed subset of $tilde{M}$".

I could not think of a proof of this claim right away or find a place where this claim is proved neither. Just wondering if this is generally true, or if I missed some conditions. How about in higher dimensions? We do not have measured foliations in higher dimensions, but is the pullback of a leaf of a minimal foliation to the universal cover always a closed subset of $tilde{M}$?

Thanks in advance!

This is indeed generally true. The proofs I know use geometry in one way or another.

Let me explain the case of a leaf $L$ of $mathcal F$ that has no singularities.

Let's fix a hyperbolic structure on $M$, and so the universal cover $tilde M$ is isometric to the hyperbolic plane $mathbb H^2$. If you understand the connection between measured foliations and geodesic laminations then you will know that corresponding to $mathcal F$ there is a geodesic lamination $mathcal L$ on $M$, there is a self-homotopy $h : M times [0,1] to M$, and there is a leaf $L'$ of $mathcal L$, such that $h mid L' times [0,1]$ is a homotopy from the identity map to $L$. It follows that in the universal covering space there are lifted leaves $tilde L$ and $tilde L'$ are properly homotopic. Since $tilde L'$ is a bi-infinite geodesic in $mathbb H^2$, it is a closed subset, hence $tilde L$ is a closed subset.

Another proof uses geometry in a different way. Using that $mathcal F$ is minimal, one can prove that there exists another measured foliation $mathcal F'$ which is transverse to $mathcal F$. The two transverse measure of $mathcal F$ and $mathcal F'$, taken together, determine a conformal structure $mu$ on $M$ and a quadratic differential $q$ of $mu$. There is an induced singular Euclidean metric on $M$: in regular local coordinates where $q = dz^2$ the metric is given by the formula $dx^2 + dy^2$ where $z=x+iy$; in singular local coordinates where $q = z^k dz^2$ ($k ge 1$), the formula is a bit more complicated. The leaf $L$ is a bi-infinite geodesic in this singular Euclidean structure. The lift to $tilde M$ of the singular Euclidean structure is a complete $text{CAT}(0)$ geodesic metric, and $tilde L$ is a complete bi-infinite geodesic in this structure, hence $tilde L$ is a closed subset.