Thematic Session on GMT 

Title: The barycentric quantitative isoperimetric inequality.

Abstract: The quantitative isoperimetric inequalities, studied since the beginning of last century, are object of a great interest in the last 20 years, also because of their several applications. In short, such an inequality aims to determine how "close" to a ball must a set be, if its perimeter is almost minimal. While the "closeness" to a ball is often considered in terms of the Fraenkel asymmetry, i.e., as the volume of the symmetric difference with respect to the best approximating ball, another possibility (actually already used 30 years ago by Fuglede) is of considering the ball centered at the barycenter of the set. This gives rise to a similar inequality, easier to prove but somehow also easier to use. In particular, it is very interesting the open question to determing the planar sets which are optimal with respect to such an inequality, and it is very close to the analogous question (studied in particular by Cicalese-Leonardi in last 10 years) with the "classical" Fraenkel asymmetry. Many of the results of this talk have been obtained in collaboration with Gambicchia (Pisa) and Torres (Barcelona).

Title: Flat interior singularities for almost-minimizers of area.

Abstract: The interior regularity of area-minimizing integral currents and semi-calibrated currents has been studied extensively in recent decades, with sharp dimension estimates and structural results established on their interior singular sets in any dimension and codimension. In stark contrast, the best result in this direction for general almost-minimizing integral currents is that due to Bombieri in the 1980’s, which demonstrates that the interior regular set is dense. We provide a construction of two types of examples that demonstrate the sharpness of Bombieri’s result, and the dramatic failure of the regularity theory developed for area-minimizing integral currents and semi-calibrated integral currents. The first example is a superposition of $C^{1,\alpha}$-graphs of a certain type with a flat singular set which can be prescribed to be any closed set with empty interior contained in a $C^{1,\alpha}$ graph. The second example is a two-dimensional almost area-minimizer in $\mathbb{R}^4$ with an accumulation of branching singularities. This is joint work with Max Goering.

Title: An elementary lemma in geometric measure theory.

Abstract: Motivated by the works of Simon and Naber and Valtorta on the rectifiability of the singular set of harmonic maps and area-minimizing integral currents in codimension 1, we consider the following general question: if a subset E of the n-dimensional Euclideal space is not H^k-\sigma-finite, does it contain a purely unrectifiable subset F with 0<H^k (F) < \infty? We show that the answer is always affirmative under very general assumptions on the set E. This provides a shortcut for some complicated covering arguments in the work of Naber and Valtorta, and in general we believe might be useful in several other contexts. Our work, which is joint with Ian Fleschler, generalizes a classical paper of Besicovitch in the fifties, which addressed the case k=1 and n=2.

Title: Isoperimetry and lower bounds on the curvature through the lens of nonsmooth geometry.

Abstract: The isoperimetric profile IX of a space X is the function that associates to each v > 0 the infimum of the perimeter of sets with volume v. In this talk I will highlight how the nonsmooth geometry essentially enters into play when proving the following smooth statement. Let n ≥ 2. For every n-dim. smooth Riemannian manifold M with Ric ≥ 0, I^{n/(n−1)}_M is concave. In the compact case "Ric ≥ 0, I^{n/(n−1)}_M is concave" is proved using the existence of isoperimetric regions and the first, and second variation of the perimeter. See, e.g., Bavard–Pansu, and Bayle. In the noncompact case isoperimetric regions might fail to exist for some volumes. In the minimization process the volume might be lost at infinity in an isoperimetric set in some possibly nonsmooth space. I will show how to recover, in the nonsmooth setting, the information given by the first, and second variation of the perimeter in the smooth setting. I will exploit this information to conclude the proof of "Ric ≥ 0, I^{n/(n−1)}_M is concave" in the noncompact case. Time allowing, I will discuss some consequences of "Ric ≥ 0, I^{n/(n−1)}_M is concave" on the isoperimetric problem. Statements similar to "Ric ≥ 0, I^{n/(n−1)}_M is concave" hold also for nonsmooth spaces with (possibly negative) synthetic lower bounds on Ric. The results presented come from several collaborations.

Title: A weighted sharp isoperimetric inequality in substatic manifolds.

Abstract: I will discuss recent advances in the geometric analysis of noncompact substatic manifolds. These spaces satisfy a curvature condition involving a potential function that vastly generalizes nonnegative Ricci curvature. Moreover, the zero-level set of such potential provides a minimal boundary for such manifolds. Substatic manifolds naturally arise as initial data sets for static spacetimes obeying Einstein's equations. I will mainly focus on a  sharp isoperimetric inequality where the volume measure is weighted with the potential, that is new also in model situations such as Schwarzschild. This is a joint work with Stefano Borghini (Trento).

Title: Optimal Quantization for Branched Optimal Transport.

Abstract: We consider the problem of best approximation of a target measure by an atomic measure with N atoms, in branched optimal transport distance. We determine the asymptotic behaviour of optimal quantizers for absolutely continuous measures as N grows to infinity, and compute the limit distribution of the corresponding point clouds. New difficulties arise, because in previously known Wasserstein semi-discrete transport results the interfaces between "basins" associated with neighboring atoms had Voronoi structure and satisfied an explicit equation, here there is no explicit description, and the interfaces have noninteger Hausdorff dimension for which no equation is known. Even proving the precise value of this dimension remains an open problem. We determine N-dependent separation distance and covering radius controls for the basins, in case the target measure is uniform. This is joint work with Paul Pegon.

Title: Hyperbolic non-smooth calculus.

Abstract: In the last 25 years, tremendous progresses have been made in the field of non-smooth analysis: after the pioneering works of the Finnish school and Cheeger’s seminal contribution, interest on the topic has been revamped by Lott-Sturm-Villani’s papers on weak lower Ricci curvature bounds. More recently, there has been a surging interest in non-smooth ‘hyperbolic’ geometry, i.e. in spaces whose smooth counterpart are Lorentzian manifolds rather than ‘elliptic’ Riemannian ones. Motivations come both from geometry and physics and concern in particular, after works of Cavalletti, McCann, Mondino, Suhr, genuinely non-smooth theories of gravity. These new geometries require new calculus tools: in this talk I will present some partial, but promising, results. Based on joint works with Beran, Braun, Calisti, McCann, Ohanyan, Rott, Saemann.

Title: Rigidity of mass-preserving 1-Lipschitz maps from integral current spaces into Euclidean space.

Abstract: We will prove that given an n-dimensional integral current space and a 1-Lipschitz map, from this space onto the n-dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, then the map has to be an isometry. We deduce as a consequence the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang--Lee--Sormani. (Joint work with Giacomo Del Nin).