The qualitative behavior for biharmonic functions on open manifolds

TL;DR

This paper investigates the properties of biharmonic functions on open manifolds with nonnegative Ricci curvature, establishing their constancy under boundedness, finite dimensionality of polynomial growth solutions, and deriving bounds for these spaces.

Contribution

It provides new results on the behavior and structure of biharmonic functions on noncompact manifolds, including finiteness and bounds of solution spaces, and extends to certain fourth-order operators in Euclidean space.

Findings

Bounded biharmonic functions are constant.

The space of polynomial growth biharmonic functions is finite dimensional.

A Weyl type bound for the space of biharmonic functions.

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.