Legendrian Gronwall conjecture

TL;DR

This paper explores an analogue of the Gronwall conjecture for Legendrian 3-webs in contact three manifolds, proposing that such webs are generally uniquely linearizable except for a specific exceptional case.

Contribution

It formulates a Legendrian version of the Gronwall conjecture, provides partial proof for maximum rank webs, and demonstrates rigidity of flat Legendrian webs under deformation.

Findings

Partial affirmation of the Legendrian Gronwall conjecture for maximum rank webs.

Sufficient flatness implies rigidity of Legendrian 3-webs under local deformation.

Identification of the exceptional case involving the Legendrian twisted cubic.

Abstract

The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjecture states that a Legendrian 3-web admits at most one distinct local linearization, with the only exception when it is locally equivalent to the dual linear Legendrian 3-web of the Legendrian twisted cubic in $\,\PP^3$. We give a partial answer to the conjecture in the affirmative for the class of Legendrian 3-webs of maximum rank. We also show that a linear Legendrian 3-web which is sufficiently flat at a reference point is rigid under local linear Legendrian deformation.