Decouplings for Surfaces of Zero Curvature

TL;DR

This paper extends the $l^2(L^p)$ decoupling theorem to all developable surfaces in three dimensions, completing the theory for zero Gaussian curvature surfaces without planar or umbilic points.

Contribution

It generalizes Bourgain-Demeter's decoupling theorem to a broader class of developable surfaces, including tangent surfaces to the moment curve.

Findings

Decoupling theorem now applies to all developable surfaces in $\,\mathbb{R}^3$

Complete the decoupling theory for zero Gaussian curvature surfaces without planar points

Includes analysis of tangent surfaces to the moment curve

Abstract

We extend the $l^2(L^p)$ decoupling theorem of Bourgain-Demeter to the full class of developable surfaces in $\mathbb{R}^3$. This completes the $l^2$ decoupling theory of the zero Gaussian curvature surfaces that lack planar (or umbilic) points. Of central interest to our study is the tangent surface associated to the moment curve.