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Why separate the complete Riemannian manifolds that admit and those that do not admit positive Green function? In summary, what is the motivation for studying parabolic and non-parabolic manifolds?

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    $\begingroup$ You may get more and better answers by adding the tag 'potential theory'. $\endgroup$
    – S.Surace
    Commented May 14, 2019 at 16:43

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One that comes to mind is that parabolicity of a Riemannian manifold is equivalent to recurrence of its Brownian motion, e.g. of the Markov process whose infinitesimal generator is $\frac12\Delta$.

See theorem 5.1 in

Grigor’yan, A. (1999). Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bulletin of the American Mathematical Society, 36(02), 135–250. https://doi.org/10.1090/S0273-0979-99-00776-4

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