Loose Legendrian embeddings in high dimensional contact manifolds

TL;DR

This paper proves an h-principle for loose Legendrian embeddings in high-dimensional contact manifolds, showing they are classified by smooth isotopy and framing, unlike in dimension 3.

Contribution

It establishes a classification of loose Legendrian embeddings via an h-principle, highlighting a fundamental difference between high and low dimensions.

Findings

Loose Legendrians have trivial pseudo-holomorphic invariants.

They are classified up to Legendrian isotopy by their smooth isotopy class and an almost complex framing.

Analogous classification results do not hold in dimension 3.

Abstract

We give an $h$--principle type result for a class of Legendrian embeddings in contact manifolds of dimension at least $5$. These Legendrians, referred to as loose, have trivial pseudo-holomorphic invariants. We demonstrate they are classified up to Legendrian isotopy by their smooth isotopy class equipped with an almost complex framing. This result is inherently high dimensional: analogous results in dimension $3$ are false.