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 {\textquoteleft}(almost) outer volume cone implies (almost) outer metric cone{\textquoteright} theorem.

}, author = {Nicola Gigli and Ivan Yuri Violo} } @booklet {2021, title = {A remark on two notions of flatness for sets in the Euclidean space}, year = {2021}, abstract = {In 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 {\textquoteleft}intrinsic{\textquoteright} 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){\textquoteright}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){\textquoteright}s behaves as the square of the numbers\ α(x,r){\textquoteright}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{\textquoteright} numbers\ β{\textquoteright}s (i.e. the one-sided version of the numbers\ α{\textquoteright}s).

}, author = {Ivan Yuri Violo} } @booklet {2021, title = {Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds}, year = {2021}, abstract = {\

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.}, author = {Francesco Nobili and Ivan Yuri Violo} }

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.