Stochastically complete manifolds and summable harmonic functions

A. A. Grigor'yan

Izvestiya. Mathematics , Tome 33 (1989) no. 2, pp. 425-432.

Résumé

Main result: if on a geodesically complete Riemannian manifold $M$ the volume $V_R$ of a geodesic ball of radius $R$ with fixed center satisfies the condition $\displaystyle\int^\infty\frac{R\,dR}{\ln V_R}=\infty$ then every nonnegative integrable superharmonic function on $M$ is equal to a constant. Bibliography: 18 titles.

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