Leaves of bounded genus Let $\mathcal{F}$ be a codimension one foliation in a closed $3$-manifold $M$. Does there exist an upper bound for the genus of the compact orientable leaves? That is, does there exist $G >0$ such that the genus of every compact orientable leaf is at most $G$?
 A: I think the answer is positive and I shall sketch a possible proof.
First of all, I claim that if we take an orientable leaf $L$ and consider a small tubular neighborhood $\mathrm{Tub}_{\epsilon}(L)$ with $L$ as it center, we can consider on $M^3$ a metric for which $L$ is totally geodesic. Since $L$ is codimension one, we can make use of the following expression for the scalar curvature of $M^3$ in this tubular neighborhood:
$$R' = R + 2\mathrm{Ricc}_{g'}(U),$$
where $R$ stands to the scalar curvature of the isometric immersion induced metric on $L$ and $U$ is normal to $L$. By continuity of the curvature, we can cantrol the scalar curvature$R^{\tilde L}$ of any other leaf $\tilde L$  in $\mathrm{Tub}_{\epsilon}(L)$ as $-\epsilon + R \leq R^{\tilde L}\leq R+\epsilon.$
Well, then we make use of Gauss--Bonnet Theorem to get
$$\int_L R' = 4\pi(2-2g) + 2\int_L\mathrm{Ricc}_{g'}(U).$$
Hence,
$$8\pi g = 8\pi -\int_LR' +2\int_L\mathrm{Ricc}_{g'}(U).$$
Hence, if $g'$ is Ricci flat we have that $g = 1$. To finish, in what follows we shall denote by the superscripts $\pm$ the positive and negative part of the underlined functions.
$$8\pi g \leq 8\pi +\int_L(R')^{-} + 2\int_L[\mathrm{Ricc}_{g'}(U)]^{+} \leq 8\pi + \int_M(R')^{-} + 2\int_M[\mathrm{Ricc}_{g'}(U)]^{+} \leq \mathrm{vol}_{g'}(M)(\max_M|R'|+2\max_M|\mathrm{Ricc}_{g'}(U)|) + 8\pi.$$
Summarily,
$$g\leq 1 + (8\pi)^{-1}\text{things that are finite due to compactness}.$$ Now
note that we have chosen a specific metric adapted to a leaf, I think compactness should imply the result by continuity on the precompact tubular neighborhood $\mathrm{Tub}_{\epsilon}(L)$. Compactness should ensure that there are at most finite bounds since there are finite number of tubular neighs.
