Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci 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.

11 pages; Criticisms are more than welcome. In the updated version, a miscited reference in the introduction has been corrected as other inaccuracies in the bibliography. In this new version assumptions H1-H2 have been removed, via a comparison argument using Ricatti's equation. The statements of the main result has been updated to reflect this. The abstract has been rewritten