A Polyharmonic Liouville Hierarchy on Complete Manifolds of Nonnegative Ricci Curvature

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.