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Ricci curvature and monotonicity for harmonic functions
http://arxiv.org/abs/1209.4669
In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green's function on open manifolds with nonnegative Ricci curvature are asymptotically umbilic.
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