Infinity behavior of bounded subharmonic functions on Ricci non-negative manifolds (Q1888703)

The author studies bounded subharmonic functions on complete Riemannian manifolds of non-negative Ricci curvature. He derives the formula \(\lim_{r\to\infty}\int_{B_x(r)} \Delta h\,dv=0\) for such functions. Further, a result by \textit{P.~Li} [Ann.\ Math.\ 124, 1--21 (1986; Zbl 0613.58032)] says that, intuitively speaking, such a function approaches its supremum at infinity when taking its average over geodesic balls. Here, the author investigates whether a pointwise limit at infinity exists. The answer is negative in general, but positive under some additional conditions on the function and the volume growth of the manifold.