A Polyharmonic Liouville Hierarchy on Complete Manifolds of Nonnegative Ricci Curvature

TL;DR

This paper develops a hierarchy of Liouville theorems for polyharmonic functions on complete manifolds with nonnegative Ricci curvature, extending classical Euclidean results and establishing new growth and constancy properties.

Contribution

It introduces a new $L^{2}$ estimate for polyharmonic functions, extending Euclidean classifications to manifolds with nonnegative Ricci curvature.

Findings

Polyharmonic functions of sublinear growth are constant.

Established a hierarchy of Liouville theorems for polyharmonic functions.

Provided the first sharp geometric extension of Euclidean polyharmonic classification.

Abstract

In this paper, we establish a complete Liouville--type hierarchy for polyharmonic functions on Riemannian manifolds with nonnegative Ricci curvature. Extending Yau's classical result for harmonic functions and our recent biharmonic Liouville theorem, we prove that on any complete manifold of nonnegative Ricci curvature, every $k$--polyharmonic function of growth $o(r^{2(k-1)})$ must in fact be $(k-1)$--polyharmonic. Iterating this procedure yields the result that all polyharmonic functions of sublinear growth are constant.The key innovation is a new $L^{2}$ estimate for the Laplacian of a polyharmonic function, obtained by induction through a delicate cutoff construction combined with a hole--filling argument. This provides the first sharp geometric extension of the Euclidean classification of polyharmonic functions to manifolds of nonnegative Ricci curvature, and completes a natural hierarchy of Yau--type Liouville theorems for iterates of the Laplacian.