10830nas a2200109 45000080041000002450068000412100065001095201041400174100001810588700002210606856009210628 2021 eng d00aMonotonicity formulas for harmonic functions in RCD(0,N) spaces0 aMonotonicity formulas for harmonic functions in RCD0N spaces3 a

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 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 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.

1 aGigli, Nicola1 aViolo, Ivan, Yuri uhttps://math.sissa.it/publication/monotonicity-formulas-harmonic-functions-rcd0n-spaces