Fourier restriction for smooth hyperbolic 2-surfaces

TL;DR

This paper establishes Fourier restriction estimates for smooth hyperbolic 2-surfaces in three-dimensional space using polynomial partitioning, leveraging hyperbolic geometry and a novel notion of transversality.

Contribution

It introduces a new approach combining polynomial partitioning with hyperbolic geometry, including a novel concept of strong transversality and the use of level set lemmas for smooth functions.

Findings

Proves Fourier restriction estimates for hyperbolic surfaces.

Develops a new framework based on hyperbolic geometry and transversality.

Utilizes level set lemmas to handle exceptional sets.

Abstract

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding "exceptional" sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.