A note on divergence of $L^{p}$-integrals of subharmonic functions and its applications

Abstract

Abstract. A non L p -integrability condition of non-constant non-negative subharmonic functions on a general complete manifold (M, g) 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 (M, g).

Parabolicity, the divergence theorem for $\delta$-subharmonic functions and applications to the Liouville theorems for harmonic maps

“…It is known that this condition on the volume growth implies the parabolicity of the manifolds (cf. [10,25]). We say that a Riemannian manifold M is parabolic or recurrent if M does not admit any nonconstant bounded subharmonic function.…”