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arXiv · 2101.03331

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

Nicola Gigli·Ivan Yuri Violo
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Abstract

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 cone implies (almost) outer metric cone' theorem.

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Nicola Gigli, Ivan Yuri Violo. 2021-01-09. Monotonicity formulas for harmonic functions in ${\rm RCD}(0,N)$ spaces. https://arxiv.org/abs/2101.03331

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