Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature

TL;DR

This paper extends Liouville's theorem to biharmonic functions on manifolds with nonnegative Ricci curvature, showing subquadratic growth functions are harmonic and sublinear growth functions are constant.

Contribution

It introduces a new local $L^2$ estimate for biharmonic functions and applies it to extend Liouville's theorem in the context of manifolds with nonnegative Ricci curvature.

Findings

Biharmonic functions of subquadratic growth are harmonic.

Biharmonic functions of sublinear growth are constant.

The theorem applies to hypersurfaces with positive sectional curvature and certain manifolds with decaying curvature.

Abstract

In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, any biharmonic function of subquadratic growth must be harmonic, and hence, any biharmonic function of sublinear growth must be constant. Our proof relies on a new local $L^2$ estimate for the Laplacian of biharmonic functions combined with a mean value inequality. Examples where our theorem applies include hypersurfaces of positive sectional curvature in $\mathbb{R}^n$, and manifolds with a pole of nonnegative Ricci curvature whose curvature decays at infinity rapidly enough.