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I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at most $d$ is finite dimensional. It is known that these $M$ have at most polynomial volume growth.

I am wondering how much the growth of $M$ is related to the dimensionality of harmonic functions. For example, is it known what happens when the volume growth is known to be not polynomial? To be more specific, suppose we ask the analogous question on spaces of negative sectional curvature: are the harmonic functions of polynomial growth there infinite dimensional? This is mainly a reference request.

Note: Edited after R W's reply below.

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For the $n$-dimensional hyperbolic space it is already the space of bounded harmonic functions that is infinitely dimensional, which follows from the integral Poisson formula.

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    $\begingroup$ By using the ball model and solving the Dirichlet problem, right? $\endgroup$
    – Sakunee
    Commented Apr 30, 2022 at 18:29
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    $\begingroup$ yes. In general harmonic functions on a complete Riemannian manifold with negative sectional curvature are identified with measures on a boundary at infinity via integration against a generalization of the Poisson kernel involving Buseman functions. It's been a while but I think the term to search for is the "Martin boundary." $\endgroup$
    – Neal
    Commented Apr 30, 2022 at 18:55
  • $\begingroup$ @Sakunee - Yes, precisely. Actually, there is no need to use the ball model when talking about the Poisson formula, sometimes the upper half-space model is more convenient. $\endgroup$
    – R W
    Commented May 1, 2022 at 9:19
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    $\begingroup$ @Neal - If you want to involve Busemann functions, then the right condition is their harmonicity, which (skipping technicalities) is quite rare unless the curvature is constant. $\endgroup$
    – R W
    Commented May 1, 2022 at 9:23

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