The qualitative behavior for biharmonic functions on open manifolds
Lin Wang, Miaomiao Zhu
TL;DR
The paper extends Yau’s Liouville-type theory from harmonic to biharmonic functions on open manifolds with nonnegative Ricci curvature and establishes a finite-dimensional, Weyl-type bound for the space of biharmonic functions with polynomial growth. It introduces a higher-order reverse Poincaré inequality under quadratic-decay Ricci conditions and leverages Li-Schoen’s mean value inequality and Cheng’s Liouville theorem to show that bounded biharmonic functions are constant, and that growth-controlled biharmonic functions exhibit rigid structure. A key outcome is the Weyl-type bound $\dim \mathcal{H}^2_{p,d}(M) \le C d^{n-1}$ for $d\ge 1$ (and $1$ for $0\le d<1$), with refined bounds on the image under the Laplacian map, aligning with Euclidean behavior. The results also include a finite-dimensional theorem for a broad class of fourth-order operators on $\mathbb{R}^n$ with coefficient conditions, highlighting the robustness of the approach beyond the biharmonic setting.
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.