Linear growth harmonic functions on complete manifolds
The main task of the author in the present paper is to prove the following theorem: `Let \(M\) be an \(n\)-dimensional complete Riemannian manifold with nonnegative Ricci curvature outside the ball \(B_p(a)\), where \(p \in M\) is a fixed point. Assume that the first Betti number of \(M\) is finite. Then the space of all linear growth harmonic functions on \(M\) has finite dimension. Moreover, if \(\text{Ric}_M \geq - K\) on \(B_p(a)\) for some nonnegative number \(K\), then there exists a constant \(C(K,a,n)\) depending only on \(K\), \(a\) and \(n\) such that the dimension is less than or equal to \(C(K,a,n)\). This result can be considered as a support to the Yau's conjecture on the dimension of the space of harmonic functions on a manifold \(M\) with nonnegative Ricci curvature.
scientific article; zbMATH DE number 870994