Some properties of subharmonic functions on complete Riemannian manifolds and their geometric applications (Q1766216)

The paper under review deals with the global behavior of subharmonic functions on a complete noncompact simply connected Riemannian manifold. In particular, the authors prove that on a strongly parabolic and strongly symmetric noncompact Riemannian manifold \((M,o)\) around \(o\) if \(u \in \mathcal C^2(M)\) is subharmonic on \(M\) with \(\Delta u \leq h\) for some \(h \in \mathcal H\) then \(u\) is harmonic on \(M,\) where \[ \mathcal H =\Bigl \{ h \in \mathcal C^0(M) \mid h \geq 0, \;i\int_1^\infty \frac{1}{V(r)} \int_{B(r)} h \,d\mu\,dr < +\infty \Bigl \} \] and \(B(r)\) denotes the geodesic ball of radius \(r\) and center \(o.\) Furthermore, under the same assumptions, if \(u \in \mathcal C^2(M)\) is a positive subharmonic function on \(M\) with \(\Delta u^\alpha \leq h\) for some \(h \in \mathcal H\) and \(\alpha \in (1,\infty),\) then \(u\) is constant on \(M.\) They also prove a comparison theorem for a noncompact Riemannian manifold with pole to be strongly parabolic. Then by using the comparison theorem for strong parabolicity, another proof of Huber's Theorem is also obtained [see \textit{A. Huber}, Comment. Math. Helv. 32, 13--72 (1957; Zbl 0080.15001)].