Harmonic Functions of Linear Growth on Kähler Manifolds with Nonnegative Ricci Curvature

The subject began in 1975, when Yau [Y1] proved that there are no nonconstant, positive harmonic functions on a complete manifold with nonnegative Ricci curvature. A few years later, Cheng [C] pointed out that using a local version of Yau’s gradient estimate, developed in his joint work with Yau [CY], one can show that there are no nonconstant harmonic functions of sublinear growth on a manifold with nonnegative Ricci curvature. Using Euclidean space as a model, this prompted Yau [Y3] to suggest in his 1981 IMU lectures the study of the space of harmonic functions of polynomial growth on manifolds with nonnegative Ricci curvature. He later conjectured that the space of harmonic functions which grows at most polynomially of order p, for p ∈ Z, on a complete manifold with nonnegative Ricci curvature is finite dimensional. Part of this conjecture was verified by Tam and the author in [TL1]. In fact, they proved a sharp estimate on the dimension of the space of harmonic functions of linear growth. To state this theorem in a precise form, let us first make the following definition: