Monotonicity Formulas for Harmonic Functions in RCD(0,N) Spaces
Gigli, N., & Violo, I. Y. (2023). Monotonicity Formulas for Harmonic Functions in RCD(0,N) Spaces. Journal of Geometric Analysis, 33(3), Article 100. https://doi.org/10.1007/s12220-022-01131-7
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Journal of Geometric AnalysisDate
2023Copyright
© The Author(s) 2022
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.
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https://converis.jyu.fi/converis/portal/detail/Publication/176981172
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Open Access funding provided by University of Jyväskylä (JYU).License
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