Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this result to prove that a complete noncompact manifold with nonnegative Ricci curvature has at least linear volume growth.
In this paper, we prove the following theorem concerning harmonic functions on these manifolds.
Theorem: Let M be a complete noncompact manifold with nonnegative Ricci curvature and at most linear volume growth. If there exists a nonconstant harmonic function, f, of polynomial growth of any given degree q, then manifold splits isometrically, M= N x R.
Harmonic functions of polynomial growth have been an object of study for some time. Until recently it was not known whether the space of harmonic functions of polynomial growth of a given degree on a manifold with nonnegative Ricci curvature was finite dimensional. Atsushi Kasue proved this result with various additional assumptions. Tobias Colding and Bill Minnicozzi have recently proven that this space is indeed finite dimensional with no additional assumptions. [CoMin] With our stronger condition of linear volume growth, we are able to prove that this space is only one dimensional directly, using a gradient estimate of Yau and previous results by the author.