A Fourier restriction theorem for a perturbed hyperbolic paraboloid

TL;DR

This paper extends Fourier restriction estimates to a specific cubic perturbation of the hyperbolic paraboloid, achieving sharp results for p > 10/3 by developing new techniques to handle the perturbation's dominant behavior.

Contribution

It introduces novel methods to analyze Fourier restriction for perturbed hyperbolic surfaces, advancing understanding beyond the classical saddle case.

Findings

Achieved sharp Fourier restriction estimates for the perturbed surface for p > 10/3.

Discovered that small-scale behavior differs significantly from the unperturbed saddle.

Developed new techniques to handle the dominant role of perturbation terms.

Abstract

In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2-surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or "saddle" z = xy. The bilinear method gave here sharp results for p > 10/3 (Lee 05, Vargas 05, Stovall 17), and this result was recently improved to p > 3.25 (Cho-Lee 17, Kim 17). This paper aims to be a first step in extending those results to more general hyperbolic surfaces. We consider a specific cubic perturbation of the saddle and obtain the sharp result, up to the end-point, for p > 10/3. In the application of the bilinear method, we show that the behavior at small scale in our surface is drastically different from the saddle. Indeed, as it turns out, in some regimes the perturbation term assumes a dominant role, which necessitates the introduction of a number of new techniques that should also be useful for the study of more general hyperbolic surfaces.