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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.

edited Mar 27 2012 at 8:50

Willie Wong
6,987●20●43

asked Mar 26 2012 at 20:58

philthyphil
11●1

What variant do you want? – Ryan Budney Mar 27 2012 at 5:52

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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 2012 at 9:25

Thanks for the suggestion. Google is my friend. – philthyphil Mar 27 2012 at 15:23

Did you find what you wanted? – Spencer May 1 2012 at 13:52

Rafe Mazzeo · Answer 1 · 2012-08-22 05:44:20Z UTC

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

answered Aug 22 at 5:44

Rafe Mazzeo
1,179●2●3

spr · Answer 2 · 2012-04-03 12:51:04Z UTC

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.

Geometric Mean Value Property

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