A transverse volume-preserving flow implies tautness in the compact cooriented three-dimensional setting

Suppose $F$ is not taut. By [F1] there is a dead-end region; take a connected component $N$ with nonempty boundary. The sign of $X$ relative to the chosen coorientation is constant on this connected region, so we may choose the transverse field $X$ co-oriented so that it points inwards along $\mathrm{\partial }N$: replacing $X$ by $-X$ preserves $L_X\mu =0$ up to sign and changes the co-orientation, so the volume-preservation hypothesis is unaffected.

A transverse flow with this co-orientation maps $N$ properly into itself: an integral curve starting in $N$ cannot cross $\mathrm{\partial }N$ outwards without violating the inward co-orientation, so ${\varphi }_t(N)\subseteq N$ for $t\ge 0$, and the inclusion is proper for every $t>0$ because the flow moves points of $N$ strictly inwards across a small collar of the boundary.

On the other hand $\mu ({\varphi }_t(N))=\mu (N)$ by [F3], and $\mu (N)<\mathrm{\infty }$ because $N$ is compact and $\mu$ is a volume form [F4]. The proper inclusion leaves in $N\setminus {\varphi }_t(N)$ a nonempty open set, hence positive measure for $t>0$, contradicting the equality of measures. Therefore no dead-end component exists and $F$ is taut.

This is Ranz's volume-preserving-flow lemma and the converse direction of Calegari's volume criterion, with completeness of the flow supplied by compactness of $M$ [F2]; the argument uses only finitely many charts and one flow, so it consumes at most the standing countable choice from [F5].