On the Harnack inequality for harmonic functions on complete Riemannian manifolds (Q1801712)

Let \(\overline M\) be a complete Riemannian manifold of dimension \(n\) with nonnegative Ricci curvature and \(M\) a complete Riemannian manifold uniformly equivalent to \(\overline M\). Suppose \(\text{Vol}(B_ x(r))\geq C_ n r^ n\), where \(B_ x(r)= \{y\in M\mid\text{dist}(y,x)\leq r\}\) and \(\text{Vol}(B_ x(r))\) the volume of \(B_ x(r)\). The author proves the Harnack inequality for harmonic functions on \(M\) and, as its applications, the Liouville theorem for harmonic functions and harmonic maps from \(M\).