A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

Abstract

AbstractWe consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ R 3 , which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method.