A sharp bilinear restriction estimate for paraboloids

TL;DR

This paper extends Wolff's sharp bilinear restriction theorem from cones to elliptic surfaces like paraboloids and spheres, improving restriction theory and addressing a conjecture.

Contribution

It adapts Wolff's argument to elliptic surfaces, providing a sharper bilinear restriction estimate for paraboloids and spheres, and resolving a conjecture.

Findings

Established a sharp bilinear restriction estimate for paraboloids and spheres

Improved upon existing restriction theory results

Partially answered a conjecture of Machedon and Klainerman

Abstract

Recently Wolff obtained a sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of ``elliptic surfaces'' such as paraboloids and spheres. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.