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

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.