Subharmonic Functions On Riemannian Manifolds
Subharmonic functions can be defined on an arbitrary Riemannian manifold.
Definition: Let M be a Riemannian manifold, and an upper semicontinuous function. Assume that for any open subset, and any harmonic function f1 on U, such that on the boundary of U, the inequality holds on all U. Then f is called subharmonic.
This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality, where is the usual Laplacian.
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