The qualitative behavior for biharmonic functions on open manifolds

Abstract

For a complete noncompact Riemannian manifold with nonnegative Ricci curvature, we show that bounded biharmonic functions are constant and the space consists of biharmonic functions with polynomial growth of a fixed rate is finite dimensional. Also, we derive a Weyl type bound for this space. Finally, we present a finite dimensional result for a class of fourth-order operators on ${\mathbb{R}}^n$ satisfying certain coefficient conditions.

This version was completed on June 18, 2025 and submitted. Comments welcome!