Decoupling for degenerate hypersurfaces

TL;DR

This paper advances decoupling theory by establishing new results for smooth hypersurfaces, showing that under certain conditions, decoupling properties extend from polynomial graphs to all smooth hypersurfaces in various dimensions.

Contribution

It proves that uniform decoupling for polynomial graphs implies decoupling for all smooth hypersurfaces, and provides unconditional decoupling results in specific dimensions.

Findings

Decoupling for all smooth hypersurfaces in $\,\mathbb R^4$ is established.

Unconditional decoupling for graphs of homogeneous polynomials in $\,\mathbb R^5$.

Conditional decoupling for general smooth hypersurfaces based on polynomial graph decoupling.

Abstract

We utilise the two principles of decoupling introduced in arXiv:2407.16108 to prove the following conditional result: assuming uniform decoupling for graphs of polynomials in all dimensions with identically zero Gaussian curvature, we can prove decoupling for all smooth hypersurfaces in all dimensions. Moreover, we are able to prove (unconditional) decoupling for all smooth hypersurfaces in $\mathbb R^4$ and graphs of homogeneous polynomials in $\mathbb R^5$.