Ricci curvature and monotonicity for harmonic functions |
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| Authors: | Tobias Holck Colding William P. Minicozzi II |
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| Affiliation: | 1. Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA
2. Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD, 21218, USA
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| Abstract: | 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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