Calculus of Variations and Geometric Measure Theory
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N. Gigli - I. Y. Violo

Monotonicity formulas for harmonic functions in ${\rm RCD}(0,N)$ spaces

created by violo on 15 Jan 2021



Inserted: 15 jan 2021
Last Updated: 15 jan 2021

Year: 2021

ArXiv: 2101.03331 PDF


We generalize to the ${\rm RCD}(0,N)$ setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative 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 AFM we also introduce the notion of electrostatic potential in ${\rm RCD}$ spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in ${\rm RCD}(K,N)$ spaces and on a new functional version of the `(almost) outer volume come implies (almost) outer metric cone' theorem.

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