On the HJY Gap Conjecture in CR geometry vs. the SOS Conjecture for polynomials

TL;DR

This paper links the HJY Gap Conjecture in CR geometry to the SOS Conjecture for polynomials, showing that the former can be derived from the latter through the CR Gauss equation and recent rigidity results.

Contribution

It establishes a novel connection between CR geometry conjectures and polynomial SOS problems, leveraging recent partial rigidity results.

Findings

HJY Gap Conjecture follows from SOS Conjecture

CR Gauss equation bridges CR mappings and polynomial sums of squares

Recent rigidity results support the conjecture connection

Abstract

We show that the Huang-Ji-Yin (HJY) Gap Conjecture concerning CR mappings between spheres follows from a conjecture regarding Sums of Squares (SOS) of polynomials. The connection between the two problems is made by the CR Gauss equation and the fact that the former conjecture follows from the latter follows from a recent result, due to the author, on partial rigidity of CR mappings of strictly pseudoconvex hypersurfaces into spheres.