Monotonicity Formulas for Harmonic Functions in RCD(0,N) Spaces
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.
Main Authors
Format
Articles
Research article
Published
2023
Series
Subjects
Publication in research information system
Publisher
Springer
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202302231879Käytä tätä linkitykseen.
Review status
Peer reviewed
ISSN
1050-6926
DOI
https://doi.org/10.1007/s12220-022-01131-7
Language
English
Published in
Journal of Geometric Analysis
Citation
- 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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Additional information about funding
Open Access funding provided by University of Jyväskylä (JYU).
Copyright© The Author(s) 2022