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

Abstract

We generalize to the $\mathrm{RCD}(0,N)$ 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 $\mathrm{RCD}$ spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in $\mathrm{RCD}(K,N)$ spaces and on a new functional version of the `(almost) outer volume cone implies (almost) outer metric cone' theorem.

Revised version

References in corpus (4)

• Well posedness of Lagrangian flows and continuity equations in metric measure spaces
• Independence on $p$ of weak upper gradients on RCD spaces
• Behaviour of the reference measure on $\mathsf{RCD}$ spaces under charts
• Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces