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Ricci curvature and monotonicity for harmonic functions

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In this paper we generalize the monotonicity formulas of “Colding (Acta Math 209:229–263, 2012)” for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., “Almgren (Preprint)”, “Colding and Minicozzi II (PNAS, 2012)”, “Garofalo and Lin (Indiana Univ Math 35:245–267, 1986)” 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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  1. Our Green’s functions will be normalized so that on Euclidean space of dimension \(n\ge 3\) the Green’s function of the Laplacian is \(r^{2-n}\).

  2. A complete manifold is nonparabolic if it admits a positive Green’s function \(G\). By a result of Varopoulos, [11], an open manifold with nonnegative Ricci curvature is nonparabolic if and only if \( \int _1^{\infty } \frac{r}{{\text{ Vol}}(B_r(x))}\,dr<\infty \). Combining the result of Varopoulos with work of Li and Yau [9], gives that \(G=G(x,\cdot )\rightarrow 0\) at infinity.

  3. If \(f:[0,\epsilon ) \rightarrow \mathbf{R}\) is continuous on \([0,\epsilon )\) and \(C^1\) on \((0,\epsilon )\), then there exist \(r_i \rightarrow 0\) with \(r_i \, f^{\prime }(r_i) \rightarrow 0\).


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The authors were partially supported by NSF Grants DMS 11040934, DMS 1206827, and NSF FRG Grants DMS 0854774 and DMS 0853501.

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Correspondence to William P. Minicozzi II.

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Communicated by L. Ambrosio.

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Colding, T.H., Minicozzi, W.P. Ricci curvature and monotonicity for harmonic functions. Calc. Var. 49, 1045–1059 (2014).

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