The h-principle fails for prelegendrians in corank 2 fat distributions

TL;DR

This paper demonstrates that the h-principle does not hold for prelegendrians in corank-2 fat distributions, revealing rigidity phenomena outside traditional contact topology and introducing new invariants and stabilization techniques.

Contribution

It provides the first examples of rigidity for maximally non-integrable distributions, develops foundational theory of prelegendrians, and introduces stabilization methods to produce loose Legendrian lifts.

Findings

Infinite family of non-isotopic prelegendrians distinguished by pseudoholomorphic invariants.

Existence of prelegendrians in all dimensions via a zooming argument.

Development of a theory including front projection and invariants robustness.

Abstract

We investigate the $h$-principle problem for fat distributions. These are maximally non-integrable distributions with natural symplectisations and contactisations, that generalize contact distributions to higher corank. We focus on the corank-$2$ case, where we study a natural class of submanifolds, which we call prelegendrians. Their key feature is that they admit a canonical Legendrian lift to the contactisation. Our main results state that the $h$-principle fails for these submanifolds in all dimensions. This is the first example of rigidity in the study of maximally non-integrable distributions, outside of contact topology. First, we find an infinite family of $(2n+1)$-tori in the standard fat $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, with the following two properties: (1) They all represent the same formal prelegendrian class, (2) but they are not prelegendrian isotopic because they are distinguished by pseudoholomorphic curve invariants of their Legendrian lift. Secondly, we define the notion of prelegendrian stabilization in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$. This allows us to take an arbitrary prelegendrian and produce another one, in the same formal class, whose Legendrian lift is loose. In order to prove these results we also develop the fundamentals of the theory of prelegendrians. This includes: (1) introducing the notion of front projection in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, (2) proving that pseudoholomorphic curve invariants are robust under perturbations of the fat structure, allowing us to transport our results to non-standard fat structures, (3) introducing a zooming argument showing that any fat structure in dimension $6$ admits prelegendrians.