Green functions for GJMS operators on spheres, Gegenbauer polynomials and rigidity theorems

Abstract

We derive explicit representation formulae of Green functions for GJMS operators on $n$-spheres, including the fractional ones. These formulae have natural geometric interpretations concerning the extrinsic geometry of the round sphere. Conversely, we discover that this special feature uniquely characterizes spheres among closed embedded hypersurfaces in $\mathbb{R}^{n+1}$. Furthermore, for $n=3,4,5$ we prove a strong rigidity theorem for Green functions of hypersurfaces in $\mathbb{R}^{n+1}$ using the Positive Mass Theorem.