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

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

1999-03-29

arxiv.org/abs/math/9903172v1

Pacific Journal of Mathematics, Vol 192, No. 1, pp 183-189, January 2000.

Mathematics

Differential Geometry

To appear in Pacific Journal of Mathematics, submitted April 1998, 7pp

Scientific paper

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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