We generalize to the RCD (0 , N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in RCD (K, N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric cone’ theorem.

Monotonicity Formulas for Harmonic Functions in RCD (0 , N) Spaces / Gigli, N.; Violo, I. Y.. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 33:3(2023), pp. 1-89. [10.1007/s12220-022-01131-7]

Monotonicity Formulas for Harmonic Functions in RCD (0 , N) Spaces

Gigli N.;Violo I. Y.
2023-01-01

Abstract

We generalize to the RCD (0 , N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in RCD (K, N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric cone’ theorem.
2023
33
3
1
89
100
https://arxiv.org/abs/2101.03331
Gigli, N.; Violo, I. Y.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135490
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