Infinity Behavior of Bounded Subharmonic Functions on Ricci Non-negative Manifolds

Abstract

In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that $\lim _{{r \to \infty }} \frac{{r^{2} }} {{V{\left( r \right)}}}{\int_{B{\left( r \right)}} {\Delta hdv{\kern 1pt} = {\kern 1pt} {\kern 1pt} 0} }$ if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.