Lower bounds on Ricci curvature limit the volumes of sets and the existenceof harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau provedthat a complete noncompact manifold with nonnegative Ricci curvature has nononconstant harmonic functions of sublinear growth. In the same paper, Yau usedthis result to prove that a complete noncompact manifold with nonnegative Riccicurvature has at least linear volume growth. In this paper, we prove thefollowing theorem concerning harmonic functions on these manifolds. Theorem: Let M be a complete noncompact manifold with nonnegative Riccicurvature and at most linear volume growth. If there exists a nonconstantharmonic function, f, of polynomial growth of any given degree q, then themanifold splits isometrically, M= N x R.
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机译：Ricci曲率的下界限制了集合的体积，并且限制了黎曼流形上的谐波函数的存在。 1975年，Shung Tung Yau证明了具有非负Ricci曲率的完全非紧流形具有次线性增长的非恒定谐波函数。在同一篇论文中，Yau使用此结果证明具有负负曲率的完整非紧致流形至少具有线性体积增长。在本文中，我们证明了关于这些流形上谐波函数的以下定理。定理：令M为具有非负曲率且最大线性增长的完整非紧流形。如果存在任意给定阶数q的多项式增长的非恒定谐波函数f，则它们的歧管等距分裂，M = N xR。
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