1999/03/29 by Christina Sormani · 1 citation
Mathematics · #math.DG #msc:53C20 #msc:31C05
published as Pacific Journal of Mathematics, Vol 192, No. 1, pp 183-189, January 2000. · To appear in Pacific Journal of Mathematics, submitted April 1998, 7pp
arxiv created 1999/03/29 · arxiv updated 2009/11/30
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 the manifold splits isometrically, M= N x R.