Tight neighborhoods of contact submanifolds

TL;DR

This paper demonstrates that small neighborhoods of contact submanifolds are tight under certain conditions and shows the absence of small positive loops of contactomorphisms in overtwisted manifolds.

Contribution

It establishes conditions for tightness of neighborhoods of contact submanifolds and links this to the non-existence of small positive loops in overtwisted manifolds.

Findings

Small neighborhoods of contact submanifolds are tight under mild normal bundle assumptions.

Overtwisted manifolds do not admit $C^0$--small positive loops of contactomorphisms.

Provides new insights into the structure of contact manifolds and their automorphisms.

Abstract

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of $C^0$--small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary.