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

Abstract

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.

D’Angelo conjecture in the third gap interval

“…The gap conjecture for the cases of k = 1, 2, 3 have been proved [Hu99,HJ01,HJY14]. Recently, P. Ebenfelt [Eb16] proposed a SOS conjecture (i.e., the Sums of Squares of Polynomial conjecture) and proved that if the SOS conjecture is true, then it implies the gap conjecture.…”