Ricci curvature and monotonicity for harmonic functions

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...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2014-03, Vol.49 (3-4), p.1045-1059
Main Authors: Colding, Tobias Holck, Minicozzi, William P.
Format: Article
Language:English
Subjects:
Analysis
Asymptotic properties
Calculus of variations
Calculus of Variations and Optimal Control
Optimization
Control
Curvature
Harmonic analysis
Harmonic functions
Manifolds
Mathematical analysis
Mathematical and Computational Physics
Mathematical problems
Mathematics
Mathematics and Statistics
Partial differential equations
Systems Theory
Theoretical
Uniqueness
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Summary: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.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-013-0610-z