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://www.math.sissa.it/publication/monotonicity-formulas-harmonic-functions-rcd0n-spaces47999nas a2200097 45000080041000002450072000412100069001135204760600182100002247788856009147810 2021 eng d00aA remark on two notions of flatness for sets in the Euclidean space0 aremark on two notions of flatness for sets in the Euclidean spac3 aIn this note we compare two ways of measuring the n-dimensional "flatness" of a set S⊂Rd, where n∈N and d>n. The first one is to consider the classical Reifenberg-flat numbers α(x,r) (x∈S, r>0), which measure the minimal scaling-invariant Hausdorff distances in Br(x) between S and n-dimensional affine subspaces of Rd. The second is an `intrinsic' approach in which we view the same set S as a metric space (endowed with the induced Euclidean distance). Then we consider numbers a(x,r)'s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at x of radius r in S and the n-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers a(x,r)'s behaves as the square of the numbers α(x,r)'s. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones' numbers β's (i.e. the one-sided version of the numbers α's).

1 aViolo, Ivan, Yuri uhttps://www.math.sissa.it/publication/remark-two-notions-flatness-sets-euclidean-space36575nas a2200109 45000080041000002450109000412100069001505203607300219100002236292700002236314856012936336 2021 eng d00aRigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds0 aRigidity and almost rigidity of Sobolev inequalities on compact 3 a

We prove that if M is a closed n-dimensional Riemannian manifold, n≥3, with Ric≥n−1 and for which the optimal constant in the critical Sobolev inequality equals the one of the n-dimensional sphere Sn, then M is isometric to Sn. An almost-rigidity result is also established, saying that if equality is almost achieved, then M is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the RCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.1 aNobili, Francesco1 aViolo, Ivan, Yuri uhttps://www.math.sissa.it/publication/rigidity-and-almost-rigidity-sobolev-inequalities-compact-spaces-lower-ricci-curvature

An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact CD space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of RCD spaces and on a Polya-Szego inequality of Euclidean-type in CD spaces.

As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the RCD-setting.