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Ivan Yuri Violo

Publications and source records attributed to Ivan Yuri Violo.

At least 19 recordsLinked to original sources
arXiv2026

Divergence-free concentrations come from vanishing sequences

A vanishing sequence $V_n$ of divergence-free matrix fields is one that is bounded in $L^1$ and is carried by open sets $A_n$ of vanishing volume. Recently established, Bouchitté's vanishing mass conjecture says that the directions such a sequence can carry are rigidly constrained: their limiting distribution must be a superposition of microstructures whose barycenters are singular matrices. We prove the converse in the non-symmetric setting: every such superposition is attained by a vanishing sequence. In fact, we are able to construct divergence-free fields $V_n$ $supported$ on $A_n$, whose relative boundary is a smooth compact manifold.

math.AP↗
arXiv2026

Mean curvature and sharp Willmore inequalities in metric spaces

The goal of this work is to introduce a notion of mean curvature for level sets of functions in non-smooth spaces with Ricci curvature bounded below, and to prove that it satisfies sharp geometric inequalities. More precisely, we define a suitable Willmore functional $\mathcal{W}$ on Sobolev functions, whose domain of finiteness is dense in $L^p$ for any $1\le p<\infty$. For any function with finite Willmore energy, we show that almost all of its level sets admit a mean curvature vector satisfying the natural integration by parts formula with respect to the tangential divergence. As a main application, we show that in ${\rm RCD}(0,N)$ spaces with Euclidean volume growth, almost every level set of the electrostatic potential possesses a mean curvature vector in the above sense. Furthermore, we prove that this vector satisfies the same sharp Willmore inequality as in the smooth setting, alongside rigidity and almost-rigidity statements. Finally, as a technical tool, we generalize the sharp isocapacitary inequality to the non-smooth setting.

math.DG↗
arXiv2026

On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 2

We characterize uniform $k$-rectifiability in Euclidean spaces in terms of a Carleson-type geometric lemma for a new notion of flatness coefficients, which we call $ι$-numbers. The characterization follows from an abstract statement about approximation by generalized planes in metric spaces, which also applies to the study of low-dimensional sets in Heisenberg groups. A key aspect is that the $ι$-coefficients are in general not pointwise comparable to the usual squared $β$-numbers for dyadic cubes on $k$-regular sets in $\mathbb{R}^n$, however our result implies that they are still equivalent in terms of a Carleson-type geometric lemma.

math.MG↗
arXiv2026

Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition

The goal of this note is to investigate quantitative stability properties of the critical Sobolev inequality in ${\sf CD}(N-1,N)$ metric measure spaces. Assuming that the optimal constant for the inequality is almost the same as the one of the round sphere, we show that the cumulative distribution of any almost extremal function is close, in Wasserstein distance, to the one of an Aubin-Talenti bubble on the round sphere. We obtain similar results for the log Sobolev inequality and the spectral gap under various curvature and dimension assumptions. In all cases we obtain a quantitative stability with sharp exponent.

math.AP↗
arXiv2025

Generalized existence of extremizers for the sharp $p$-Sobolev inequality on Riemannian manifolds with nonnegative curvature

We study the generalized existence of extremizers for the sharp $p$-Sobolev inequality on noncompact Riemannian manifolds in connection with nonnegative curvature and Euclidean volume growth assumptions. Assuming a nonnegative Ricci curvature lower bound, we show that almost extremal functions are close in gradient norm to radial Euclidean bubbles. In the case of nonnegative sectional curvature lower bounds, we additionally deduce that vanishing is the only possible behavior, in the sense that almost extremal functions are almost zero globally. Our arguments rely on nonsmooth concentration compactness methods and Mosco-convergence results for the Cheeger energy on noncompact varying spaces, generalized to every exponent $p\in (1,\infty)$.

math.AP↗
arXiv2025

Regularity for quasilinear elliptic equations in metric measure spaces

In the present article we prove second-order and Lipschitz regularity for quasilinear elliptic equations in metric spaces endowed with a lower bound on the Ricci curvature. The estimates we obtain are quantitative and cover a large class of elliptic equations with polynomial growth. As a particular case we settle the Lipschitz regularity of $p$-harmonic functions for all values of $p\in(1,\infty)$, proving also a Cheng-Yau type inequality. These results are the first in this setting that simultaneously address a wide family of elliptic operators and extend beyond the classical Hölder regularity theory. Our strategy rests on the use of Galerkin's method, which we employ as an alternative to the traditional difference quotients technique.

math.AP↗
arXiv2025

Fine Pólya-Szegő rearrangement inequalities in metric spaces and applications

We study fine Pólya-Szegő rearrangement inequalities into weighted intervals for Sobolev functions and functions of bounded variation defined on metric measure spaces supporting an isoperimetric inequality. We then specialize this theory to spaces with synthetic Ricci lower bounds and characterize equality cases under minimal assumptions. As applications of our theory, we show new results around geometric and functional inequalities under Ricci lower bounds answering also questions raised in the literature. Finally, we study further settings and deduce a Faber-Krahn theorem on Euclidean spaces with radial log-convex densities, a boosted Pólya-Szegő inequality with asymmetry reminder on weighted convex cones, the rigidity of Sobolev inequalities on Euclidean spaces outside a convex set and a general lower bound for Neumann eigenvalues on open sets in metric spaces.

math.AP↗
arXiv2025

Second-order estimates for the $p$-Laplacian in RCD spaces

We establish quantitative second-order Sobolev regularity for functions having a $2$-integrable $p$-Laplacian in bounded RCD spaces, with $p$ in a suitable range. In the finite-dimensional case, we also obtain Lipschitz regularity under the assumption that $p$-Laplacian is sufficiently integrable. Our results cover both $p$-Laplacian eigenfunctions and $p$-harmonic functions having relatively compact level sets.

math.MG↗
arXiv2025

On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 1

We introduce new flatness coefficients, which we call $ι$-numbers, for Ahlfors $k$-regular sets in metric spaces ($k\in \mathbb{N}$). Using these coefficients for $k=1$, we characterize uniform $1$-rectifiability in rather general metric spaces, completing earlier work by Hahlomaa and Schul. Our proof proceeds by quantifying an isometric embedding theorem due to Menger, and by an abstract argument that allows to pass from a local covering by continua to a global covering by $1$-regular connected sets.

math.MG↗
arXiv2024

Notes on the Cheeger and Colding version of the Reifenberg theorem for metric spaces

The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological Hölder manifold. Remarkably, this generalizes to metric spaces, where the approximation by planes is replaced by the Gromov-Hausdorff distance. This fact was shown by Cheeger and Colding in an appendix of one of their celebrated works on Ricci limit spaces [8]. Given the recent interest around this statement in the growing field of analysis in metric spaces, in this note we provide a self contained and detailed proof of the Cheeger and Colding result. Our presentation substantially expands the arguments in [8] and makes explicit all the relevant estimates and constructions. As a byproduct we also shows a biLipschitz version of this result which, even if folklore among experts, was not present in the literature. This work is an extract from the doctoral dissertation of the second author.

math.MG↗
arXiv2024

Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal function of the round sphere. In the setting of non-negative Ricci curvature and Euclidean volume growth, we show an analogous result in comparison with the extremal functions in the Euclidean Sobolev inequality. As an application, we deduce a stability result for minimizing Yamabe metrics. The arguments rely on a generalized Lions' concentration compactness on varying spaces and on rigidity results of Sobolev inequalities on singular spaces.

math.DG↗
arXiv2023

Ricci curvature bounded below and uniform rectifiability

We prove that Ahlfors-regular RCD spaces are uniformly rectifiable. The same is shown for Ahlfors regular boundaries of non-collapsed RCD spaces. As an application we deduce a type of quantitative differentiation for Lipschitz functions on these spaces.

math.MG↗
arXiv2023

Pleijel nodal domain theorem in non-smooth setting

We prove the Pleijel theorem in non-collapsed RCD spaces, providing an asymptotic upper bound on the number of nodal domains of Laplacian eigenfunctions. As a consequence, we obtain that the Courant nodal domain theorem holds except at most for a finite number of eigenvalues. More in general, we show that the same result is valid for Neumann (resp. Dirichlet) eigenfunctions on uniform domains (resp. bounded open sets). This is new even in the Euclidean space, where the Pleijel theorem in the Neumann case was open under low boundary-regularity.

math.SP↗
arXiv2023

Topological regularity of isoperimetric sets in PI spaces having a deformation property

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm{RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm{RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.

math.MG↗
arXiv2023

Fat triangles inscribed in arbitrary planar domains

In 1964 A. Bruckner observed that any bounded open set in the plane has an inscribed triangle, that is a triangle contained in the open set and with the vertices lying on the boundary. We prove that this triangle can be taken uniformly fat, more precisely having all internal angles larger than $\sim 0.3$ degrees, independently of the choice of the initial open set. We also build a polygon in which all the inscribed triangles are not too-fat, meaning that at least one angle is less than $\sim 55$ degrees. These results show the existence of a maximal number $Θ$ strictly between 0 and 60, whose exact value remains unknown, for which all bounded open sets admit an inscribed triangle with all angles larger than or equal to $Θ$ degrees.

math.MG↗
arXiv2022

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

We prove that if $M$ is a closed $n$-dimensional Riemannian manifold, $n \ge 3$, with ${\rm Ric}\ge n-1$ and for which the optimal constant in the critical Sobolev inequality equals the one of the $n$-dimensional sphere $\mathbb{S}^n$, then $M$ is isometric to $\mathbb{S}^n$. 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 ${\rm RCD}$-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact ${\rm CD}$ 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 ${\rm RCD}$ spaces and on a Polya-Szego inequality of Euclidean-type in ${\rm CD}$ 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 ${\rm RCD}$-setting.

math.DG↗
arXiv2021

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

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

math.DG↗