Linearization and Gronwall conjecture of Legendrian webs

Abstract

The Gronwall conjecture, which is still open, asserts that if a 3-web in the plane is linearizable, then the linearization is unique modulo projective transformations. We prove the conjecture for Legendrian d-webs, provided d ≥ 4. Precisely, our theorem states that, if a Legendrian d-web with d ≥ 4 in the (real or complex) 3-dimensional contact manifold is linearizable, then there is a unique linearization of the Legendrian d-web up to a contact projective transformation. For the proof, we use the linearization technique of the third order ordinary differential equations and the Schwarzian derivatives of contact transformations.