Decoupling for Ruled Hypersurfaces Generated by a Curve

TL;DR

This paper generalizes decoupling techniques for ruled hypersurfaces generated by curves to higher dimensions, revealing new geometric insights and achieving optimal decoupling bounds near curves.

Contribution

It extends decoupling theory to higher-dimensional ruled hypersurfaces, introducing a new approach to handle additional rulings and achieving optimal $\, ext{L}^p$ decoupling.

Findings

Successful extension of decoupling to higher dimensions

Case-by-case analysis of rulings reveals geometric properties

Achieves optimal $\, ext{L}^p$ decoupling near curves

Abstract

We extend previous work on the two-dimensional developable tangent surface to its higher dimensional analogues $\mathfrak{M} \subset \mathbb{R}^{n+1}$. The approach here similarly applies cylindrical approximate decoupling at its core, albeit in a new format. However, the presence of additional rulings as $n$ increases necessitates a case-by-case analysis, which in itself reveals interesting aspects of the geometry of $\mathfrak{M}$. The contributions of this paper can be viewed as culminating in the optimal $\ell^2(L^p)$ decoupling over Frenet boxes approximating a suitably defined, arbitrarily thin neighborhood of a curve $\phi$.