Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

TL;DR

This paper develops decoupling theory for hypersurfaces with nonzero Gaussian curvature, leading to sharp Strichartz estimates for hyperbolic Schrödinger equations and implications for Vinogradov's mean value theorem.

Contribution

It extends decoupling theory to hypersurfaces with nonzero Gaussian curvature and applies it to hyperbolic Schrödinger equations and number theory.

Findings

Established decoupling for hypersurfaces with nonzero Gaussian curvature.

Derived sharp Strichartz estimates for hyperbolic Schrödinger equations.

Proved $l^2$ decoupling for non-degenerate curves with applications to Vinogradov's mean value theorem.

Abstract

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from \cite{BD3}. As a consequence of this we obtain sharp (up to $\epsilon$ losses) Strichartz estimates for the hyperbolic Schr\"odinger equation on the torus. Our second main result is an $l^2$ decoupling for non degenerate curves which has implications for Vinogradov's mean value theorem.