Rigid Flat Webs on the Projective Plane

TL;DR

This paper investigates flat webs on the projective plane with zero curvature, utilizing local regularity criteria and the Legendre transform to classify and construct infinite families of such webs.

Contribution

It introduces a criterion for curvature regularity near discriminants and demonstrates that Legendre transforms of reduced convex foliations produce zero curvature webs, revealing new classes of flat webs.

Findings

Legendre transform links convex foliations to zero curvature webs

Infinite families of convex foliations yield non-deformable flat webs

Criteria for curvature regularity at discriminant neighborhoods

Abstract

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Legendre transform, is an avatar of classical projective duality in the realm of differential equations. We show that the Legendre transform of what we call reduced convex foliations are webs with zero curvature, and we exhibit a countable infinity family of convex foliations which give rise to a family of webs with zero curvature not admitting non-trivial deformations with zero curvature.