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Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic operator applied to a function, u, integrated over geodesic balls, to control the size of u. Locally is enough; I don't care about issues involving caustics, et cetera.

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What variant do you want? – Ryan Budney Mar 27 '12 at 5:52
It seems to me that if you google "mean value property harmonic functions riemannian manifold" you get lots of useful references. – Deane Yang Mar 27 '12 at 9:25
Thanks for the suggestion. Google is my friend. – user22425 Mar 27 '12 at 15:23
Did you find what you wanted? – Spencer May 1 '12 at 13:52

Schoen & Yau, Lectures on Differential Geometry (International Press, 1994), Chapter II, Section 6: Mean value inequality for subharmonic functions. This is what you want.

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You may try:

(1) Grigorʹyan, A. A. Stochastically complete manifolds and summable harmonic functions. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 5, 1102--1108, 1120; translation in Math. USSR-Izv. 33 (1989), no. 2, 425–432

(2) Leon Karp, Subharmonic functions, harmonic mappings and isometric immersions (pp. 133--142); in Seminar on Differential Geometry. Papers presented at seminars held during the academic year 1979–1980. Edited by Shing Tung Yau. Annals of Mathematics Studies, 102. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.

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