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

TL;DR

This work proves that the $h$-principle fails for prelegendrian submanifolds in corank-$2$ fat distributions, signaling rigidity beyond classical contact topology. The authors develop a comprehensive framework—fat manifolds, contactisations, and prelegendrian fronts—and introduce co-real spinning and prelegendrian stabilization to generate and distinguish infinite families of prelegendrian tori that are formally isotopic but not prelegendrian isotopic, as detected by Legendrian invariants like Legendrian contact homology. They also show that stabilization can yield loose lifts and that exotic prelegendrians exist in multiple corank-$2 fat manifolds, including standard and non-standard fat structures. The results significantly extend rigidity phenomena from contact topology to higher-corank fat geometries, offering new tools (nilpotentisation, front projections, and stabilization techniques) and raising questions about broader applicability and higher corank cases. The study provides a pathway to understand when formal data fails to classify geometric submanifolds in fat manifolds, with potential implications for symplectic/Legendrian invariants in geometric analysis and sub-Riemannian geometry.

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.

Figures (5)

• Figure 1: Left: the front of the cusp model $L^0_\mathrm{cusp}$. Right: the stabilized front $L_\mathrm{loose}$.
• Figure 2: A schematic depiction of the front spinning of the Legendrian unknot $U\subseteq (J^1\mathbb{R},\xi_\mathrm{std})$ along the standard circle $\mathbb{S}^1\subseteq \mathbb{R}^2$.
• Figure 3: From left to right the fronts $\hat{\mathcal{F}}_s\subseteq J^0\mathbb{R}$, $s\in[0,1]$, of the $1$-dimensional Reidemeister moves I, II and III. The higher dimensional moves are locally modelled on $\hat{\mathcal{F}}_s\times\mathbb{R}^n\subseteq J^0\mathbb{R}\times\mathbb{R}^n=J^0\mathbb{R}^{n+1}$, $s\in[0,1]$.
• Figure 4: The Legendrian $W_s$, where $s$ is the number of crossings in the right part.
• Figure 5: A formal prelegendrian isotopy between of $\Lambda_s=\mathrm{prespin}_{\hat{e}} [W_s]$ and $\mathrm{prespin}_{\hat{e}}[U]$, where $U\subseteq (J^1\mathbb{R},\xi_\mathrm{std})$ is the standard Legendrian unknot. The first and fourth arrow indicate prelegendrian RII moves, and the last one several prelegendrian RI moves along parallel copies of the co-real $\mathbb{T}^{2n}$. The second and third arrow describe a prelegendrian $N$-pushing along the co-real $\mathbb{T}^{2n}$.

Theorems & Definitions (139)

• definition 1.1
• remark 1.2
• remark 1.3
• Lemma 1.4
• definition 1.5
• remark 1.6
• remark 1.7
• remark 1.8
• Theorem 1.9
• Theorem 1.10