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Harmonic Functions on Manifolds with Nonnegative Ricci Curvature and Linear Volume Growth

Christina Sormani
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Abstract

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.

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Christina Sormani. 1999-03-29. Harmonic Functions on Manifolds with Nonnegative Ricci Curvature and Linear Volume Growth. https://arxiv.org/abs/math/9903172

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