Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature

TL;DR

The paper proves a Liouville-type result for biharmonic functions on complete Riemannian manifolds with nonnegative Ricci curvature and a pole: any biharmonic function of subquadratic growth is harmonic, and thus any biharmonic function of sublinear growth is constant. The authors develop a new local $L^2$ estimate for $\Delta u$, pair it with a hole-filling iteration, and use a mean value inequality for $\Delta u$ to deduce harmonicity. The argument reduces to Yau’s classical Liouville theorem once $\Delta u$ is shown to vanish. The results provide a sharp higher-order analogue of Yau’s theorem and have potential applications to hypersurfaces of positive curvature and manifolds with decaying curvature at infinity.

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.

Theorems & Definitions (12)

• Theorem 1
• Lemma 1: Cutoff with explicit bounds
• proof
• Corollary 2
• proof
• Lemma 3: Hole-filling iteration
• proof
• Lemma 4: Mean value inequality for $u^2$
• Lemma 5
• proof