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.
In Memory of Professor Shoshichi Kobayashi