On the hyperbolicity criterion for noncompact Riemannian manifolds of special type
An \(n\)-dimensional Riemannian manifold is strongly parabolic if any positive harmonic function on it is a constant. The manifold is parabolic if the set of positive harmonic functions on it is of finite dimension. All other Riemannian manifolds are non-parabolic (or hyperbolic). Let \({\mathcal M}\) be a complete \(n\)-dimensional Riemannian manifold that can be represented as the union \({\mathcal M=B\cup D}\) where \({\mathcal B}\) is compact and \({\mathcal D}\) is isometric to the direct product \({\mathbb{R}_+}\times{\mathcal S}\) (where \({\mathbb{R}}_+=(0,\infty )\) and \({\mathcal S}\) is a compact Riemannian manifold), with the metric \(ds^2= h^2(r)dr^2+g^2(r)d\theta ^2\), where \(h(r), g(r)\) are smooth positive functions on \({\mathbb{R}}_+\) and \(d\theta^2 \) is a Riemannian metric on \({\mathcal S}\). Consider the integral \(I=\int_{r_0}^\infty h(t) g^{1-n}(t) (\int _{r_0}^\infty h(\xi ) g^{n-3}(\xi)d\xi )dt\) where \(r_0=\)const\(>0\). Theorem. Assume that \(I=\infty\). Then \({\mathcal M}\) is strongly parabolic. 2. Assume that \(I<\infty\). Then \({\mathcal M}\) is hyperbolic. Moreover, for any function \(\Phi (\theta )\), continuous on \({\mathcal S}\), there is a harmonic function \(u\) such that \(\lim_{r\to \infty}u(r,\theta )=\Phi (\theta )\).
scientific article; zbMATH DE number 1028542