Existence and classification of maximal growth distributions

Abstract

This article tackles the problem of the existence and classification of maximal growth distributions on smooth manifolds. We show that maximal growth distributions of $\text{rank}>2$ abide by a full $h$-principle in all dimensions. We make use of M. Gromov’s higher order convex integration and, on the way, we establish a new criterion for checking ampleness of a differential relation.
As a consequence, we answer in the positive, for $k>2$, the long-standing open question posed by M. Kazarian and B. Shapiro more than 25 years ago about whether any parallelizable manifold admits a $k$-rank distribution of maximal growth. We also answer several related open questions.
For completeness, we show that the differential relation of maximal growth for rank-$2$ distributions is not ample in any ambient dimension. Non-ampleness of the Engel and the $(2,3,5)$-conditions follow as particular cases.