A note on divergence of -integrals of subharmonic functions and its applications

A non -integrability condition of non-constant non-negative subharmonic functions on a general complete manifold is given in an optimal form. As an application in differential geometry, several topics related to parabolicity of manifolds, the Liouville theorem for harmonic maps and conformal deformation of metrics are shown without any assumption on the Ricci curvature of .