Slicing surfaces and Fourier restriction conjecture

TL;DR

This paper explores the Fourier restriction problem, establishing implications between conjectures for various surfaces and proving new estimates for surfaces in three-dimensional space that are locally planar and finite type.

Contribution

It demonstrates that restriction conjectures for certain surfaces imply those for the cone and introduces a novel restriction estimate for specific surfaces in three dimensions.

Findings

Restriction conjectures for sphere, paraboloid, elliptic hyperboloid imply those for the cone.

New restriction estimate for surfaces in R^3 locally isometric to the plane.

Results connect restriction phenomena across different surfaces.

Abstract

We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in $\mathbb{R}^n$ implies that for the cone in $\mathbb{R}^{n+1}$. We also prove a new restriction estimate for any surface in $\mathbb{R}^3$ locally isometric to the plane and of finite type.