Harmonic functions on Riemannian manifolds

Summary

A harmonic function on a Riemannian manifold is a function killed by the Laplace–Beltrami operator of the metric. This extends the classical harmonic functions of Euclidean space to curved spaces. Their behaviour depends on the geometry: on a connected closed manifold they are constant. On a complete manifold with nonnegative Ricci curvature, the positive ones are constant. They link analysis with geometry, through gradient estimates, Liouville theorems and the study of the manifold at infinity.