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A Riemannian manifold is called biparabolic if any positive bisuperharmonic function is harmonic. We provide a criterion of biparabolicity in terms of the Green function of the Laplacian, as well as a sharp sufficient condition for biparabolicity in terms of the volume growth function.
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Shokoufe Faraji, Alexander Grigor'yan, On biparabolicity of Riemannian manifolds. Rev. Mat. Iberoam. 35 (2019), no. 7, pp. 2025–2034DOI 10.4171/RMI/1108