Ricci curvature and monotonicity for harmonic functions

By T. Colding, W. Minicozzi. 2012

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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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/b4fd572831b9ee0182d61363da7f79bd4dcfca9b

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In Applied Mathematics

Citations in the corpus

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature (2018, methodology): “ in position to complete the proof of Theorem 3.2. Conclusion of the proof of Theorem 3.2. We are going to pass to the limit as S→ ∞ in (3.28). The following argument is due to Colding and Minicozzi, [26]. We ﬁrst prove that the derivative of Ψβ has a sign. By (3.28), we have, for β≥ 1 and for any 0”
On the 1/H-flow by p-Laplace approximation: New estimates via fake distances under Ricci lower bounds (2019, background): “In the linear case, similar identities were obtained in [24] and in [3, 13].”