Polynomials with small norm on compact Riemannian homogeneous manifolds

TL;DR

This paper investigates the existence of polynomials with small norms on compact Riemannian homogeneous manifolds, addressing challenges posed by unbounded eigenfunctions and introducing a geometric inequality-based method.

Contribution

It introduces a novel method using geometric inequalities to study small norm polynomials on manifolds with unbounded eigenfunctions.

Findings

Established existence results for small norm polynomials on certain manifolds.

Developed a new geometric inequality approach applicable to non-Abelian settings.

Extended classical results from the circle to more general homogeneous manifolds.

Abstract

We consider the problem of existence of polynomials with small norm. This range of problems has been extensively studied by many authors in the case of the unit circle (or a compact Abelian group), i.e. when the characters are bounded. In general, on compact homogeneous Riemannian manifolds, the eigenfunctions of the Laplace-Beltrami operator are not uniformly bounded. This creates difficulties of a fundamental nature in applications of known methods and results. The method we develop is based on a geometric inequality between norms induced by two convex bodies in Euclidean space.