arXiv subjects

Gioacchino Antonelli

Publications and source records attributed to Gioacchino Antonelli.

At least 19 recordsLinked to original sources

arXiv2026

Removability of non-isolated singularities for Einstein metrics and RCD spaces

In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $> 3- \frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^\infty$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,α}$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^\infty$ and sufficiently close to a smooth background metric.

math.DG↗

arXiv2026

Universal Volume Growth Bounds from Positive Intermediate Curvature

Let $n,m$ be integers such that $n\geq2$ and $0\leq m\leq n-2$. Let $C_{m+1}$ denote the $(m+1)$-intermediate curvature introduced by Brendle--Hirsch--Johne. We prove that there are constants $ν(n,m),C(n,m)>0$ such that the following holds. If $(M^n,g)$ is complete and connected and, for $δ\geq 0$, \[ \mathrm{Ric}\geq-δ^2, \qquad C_{m+1}\geq 1, \] then \[ δR\leqν(n,m) \quad\Longrightarrow\quad \mathrm{Vol} B_R(p) \leq C(n,m)R^m \quad \text{for every $p\in M$ and $R>0$.} \] In particular, taking $m=n-2$ and $δ=0$ gives Gromov's conjectured codimension-two volume growth estimate under $\mathrm{Ric} \geq0$ and $\mathrm{Scal} \geq1$.

math.DG↗

arXiv2026

Planar lamplighter is not of negative type

The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

math.MG↗

arXiv2026

Spaces of metrics with positive spectral scalar curvature

Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^γ(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-γΔ_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $γ>0$, or if $n\ge3$ and $0< γ\leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\hookrightarrow R^γ(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. Then, we prove that if $n\ge3$ and $γ>4(n-1)/(n-2)$, the space $R^γ(M)$ is contractible, and hence nonempty. This solves a homotopy-theoretic strengthening of a conjecture of Gromov (Conjecture 3, Section 6.1.2, "Four Lectures on Scalar Curvature") in the maximal possible coefficient range. Concerning Gromov's conjecture we also treat the equivariant case and the case of manifolds with boundary.

math.DG↗

arXiv2026

Ergodic maps and the cohomology of nilpotent Lie groups

In this paper, we study how the cohomology of nilpotent groups is affected by Lipschitz maps. We show that, given a smooth Lipschitz map $f$ between two simply-connected nilpotent Lie groups $G$ and $H$, there is a map $ψ$ that induces an ergodic measure on the space of functions from $G$ to $H$. We call such maps ergodic maps. We show that when $ψ$ is an ergodic map, the pullback $ψ^*ω$ of a differential form $ω$ admits a well-defined amenable average $\overline{ψ^{*}}ω$, and $\overline{ψ^*}$ is a homomorphism of cohomology algebras. In the case that $f$ is a quasi-isometry, the ergodic map $ψ$ is also a quasi-isometry, and $\overline{ψ^*}$ is an isomorphism. This lets us generalize and provide a simplified, self-contained proof of the theorem due to Shalom, Sauer, and Gotfredsen--Kyed that quasi-isometric nilpotent groups have isomorphic cohomology algebras.

math.GR↗

arXiv2026

Vertical curves and vertical fibers in the Heisenberg group

Let $\mathbb{H}$ denote the three-dimensional Heisenberg group. In this paper, we study vertical curves in $\mathbb{H}$ and fibers of maps $\mathbb{H} \to \mathbb{R}^2$ from a metric perspective. We say that a set in $\mathbb{H}$ is a vertical curve if it satisfies a cone condition with respect to a homogeneous cone with axis $\langle Z \rangle$, the center of $\mathbb{H}$. This is analogous to the cone condition used to define intrinsic Lipschitz graphs. In the first part of the paper, we prove that connected vertical curves are locally biHölder equivalent to intervals. We also show that the class of vertical curves coincides with the class of intersections of intrinsic Lipschitz graphs satisfying a transversality condition. Unlike intrinsic Lipschitz graphs, the Hausdorff dimension of a vertical curve can vary; we construct vertical curves with Hausdorff dimension either strictly larger or strictly smaller than 2. Consequently, there are intersections of intrinsic Lipschitz graphs with Hausdorff dimension either strictly larger or strictly smaller than 2. In the second part of the paper, we consider smooth functions $β$ from the unit ball $B$ in $\mathbb{H}$ to $\mathbb{R}^2$. We show that, in contrast to the situation in Euclidean space, there are maps such that $β$ is arbitrarily close to the projection $π$ from $\mathbb{H}$ to the horizontal plane, but the average $H^2$ measure of a fiber of $β$ in $B$ is arbitrarily small.

math.MG↗

arXiv2026

Isoperimetric problems and lower bounds on curvature

This note surveys some classical results and recent developments on the interplay between lower curvature bounds and the isoperimetric problem. It is based on mini-courses given at the "European Doctorate School of Differential Geometry" (Granada, July 2024) and the summer school "Optimal transport, heat flow and synthetic Ricci bounds" (Chicago, June 2025).

math.DG↗

arXiv2026

New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry

We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\geq3$ satisfies $$ λ_1\left(-\frac{n-1}{n-2}Δ+\mathrm{Ric}\right)\geq n-1, $$ then $\operatorname{vol}(M)\leq\operatorname{vol}(\mathbb S^{n})$, and $π_1(M)$ is finite. The constant $\frac{n-1}{n-2}$ cannot be improved, and if $\mathrm{vol}(M)=\mathrm{vol}(\mathbb S^n)$ holds, then $M\cong \mathbb S^{n}$. A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem, and unequally warped $μ$-bubbles. As an application, in dimensions $3\leq n\leq 5$, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive spectral biRicci curvature. Furthermore, the main result of this paper is applied in Mazet's recent solution of the stable Bernstein problem in $\mathbb R^6$.

math.DG↗

arXiv2026

Connected sum of manifolds with spectral Ricci lower bounds

Let $n > 2$, $γ> \frac{n-1}{n-2}$, and $λ\in \mathbb{R}$. We prove that if $M$ and $N$ are two smooth $n$-manifolds that admit a complete Riemannian metric satisfying \[-γΔ+ \mathrm{Ric} > λ,\]then the connected sum $M \# N$ also admits such a metric. The construction geometrically resembles a Gromov-Lawson tunnel; the range $ γ> \frac{n-1}{n-2} $ is sharp for this to hold.

math.DG↗

arXiv2026

Area of Hölder curves and coarea formula on the Heisenberg group

We prove the coarea formula for Lipschitz maps from the subriemannian $n$th Heisenberg group $\mathbb H_n$ to $\mathbb R^{2n}$. Our result is new even when $n=1$ and provides the simplest vector-valued instance of the coarea formula in subriemannian geometry. This answers a question left open in the works of Magnani, Kozhevnikov, Magnani--Stepanov--Trevisan, and Julia--Nicolussi Golo--Vittone. The main difficulty of the proof is that a fiber of a $C^1_{\mathrm{H}}$ map $f: \mathbb H_n\to \mathbb R^{2n}$ is typically an unrectifiable curve. Its measure depends on the symplectic area of its projection to $\mathbb R^{2n}$. A bound on this area would imply the coarea formula, but examples of Kozhevnikov show that this area can be infinite or undefined. To overcome this, we introduce an integral that we use to define both the symplectic area of $\frac{1}{2}$--Hölder curves in $\mathbb R^{2n}$ and the symplectic area of projections of vertical curves in $\mathbb H_n$. Then, we give a geometric condition for this integral to converge. This yields, in addition, new results on the existence of the signed area of $\tfrac12$--Hölder planar curves that may be of independent interest. Finally, we use $β$--number estimates from the Fässler--Orponen Dorronsoro Theorem to show that this geometric condition holds for almost every fiber.

math.MG↗

arXiv2026

Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary

We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the conditions \[ λ_1(-γΔ+\mathrm{Ric})\geq 0,\qquad H_{\partial M}\geq 0, \] and in the sharp range $0\leq γ<4$ if $n=2$, and $0\leqγ<\frac{n-1}{n-2}$ if $n\geq3$, a (possibly noncompact) complete manifold with disconnected boundary, with at least one compact boundary component, must split isometrically as a product $[0,L]\times Σ$. Our second main contribution is a topological rigidity result for the relative fundamental group $π_1(M,\partial M)$, combined with a deep theorem of Lawson--Michelsohn. We prove that, in dimensions $n\neq4$, any compact manifold with boundary satisfying the two inequalities above, with at least one of them strict, admits a metric with positive sectional curvature and strictly mean-convex boundary, provided $γ\geq0$ if $n=2$, and $0\leqγ\leq\frac{n-1}{n-2}$ if $n\geq3$. This range of $γ$ is sharp for the latter result to hold.

math.DG↗

arXiv2026

Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

This paper deals with quasi-local isoperimetric versions of the positive mass theorem on $3$-manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense. As a corollary, we derive existence results for isoperimetric sets in such low regularity setting. Our main tool is a new local version of the weak inverse mean curvature flow enjoying $C^0$-stable quantitative estimates.

math.DG↗

arXiv2025

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces. In this setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov--Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray. Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and we characterize the geometric rigidity at the level of the isoperimetric profile. Specializing our study to the $2$-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary. The proofs exploit a variational approach, and in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.

math.DG↗

arXiv2025

Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset (0,\infty)$ with density $1$ at infinity such that for every $V\in \mathcal{G}$ there is a unique isoperimetric set of volume $V$ in $M$; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals $I_n\subset (0,\infty)$ such that isoperimetric sets with volumes $V\in I_n$ exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.

math.DG↗

arXiv2024

A sharp spectral splitting theorem

We prove a sharp spectral generalization of the Cheeger--Gromoll splitting theorem. We show that if a complete non-compact Riemannian manifold $M$ of dimension $n\geq 2$ has at least two ends and \[ λ_1(-γΔ+\mathrm{Ric})\geq 0, \] for some $γ<\frac{4}{n-1}$, then $M$ splits isometrically as $\mathbb R\times N$ for some compact manifold $N$ with nonnegative Ricci curvature. We show that the constant $\frac{4}{n-1}$ is sharp, and the multiple-end assumption is necessary for any $γ>0$.

math.DG↗

arXiv2024

Sharp defective log-Sobolev inequalities on H-type groups

In this paper we prove a sharp defective log-Sobolev inequality on H-type groups. Then we use such an inequality to show exponential integrability of Lipschitz functions with respect to the heat kernel measure. A defective log-Sobolev-type inequality for the Gaussian-like measure with respect to the sub-Riemannian distance is also proved on arbitrary H-type groups.

math.AP↗

arXiv2024

Carnot rectifiability and Alberti representations

A metric measure space is said to be Carnot-rectifiable if it can be covered up to a null set by countably many biLipschitz images of compact sets of a fixed Carnot group. In this paper, we give several characterisations of such notion of rectifiability both in terms of Alberti representations of the measure and in terms of differentiability of Lipschitz maps with values in Carnot groups. In order to obtain this characterisation, we develop and study the analogue of the notion of Lipschitz differentiability space by Cheeger, using Carnot groups and Pansu derivatives as models. We call such metric measure spaces Pansu differentiability spaces (PDS).

math.MG↗

arXiv2024

Nonexistence of isoperimetric sets in spaces of positive curvature

For every $d\ge 3$, we construct a noncompact smooth $d$-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below $1$. We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in $\mathbb R^d$. The examples we construct have nondegenerate asymptotic cone. The dimensional constraint $d\ge 3$ is sharp. Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed in nonnegatively curved spaces with nondegenerate asymptotic cones isoperimetric sets with large volumes always exist. This is the first instance of noncollapsed nonnegatively curved space without isoperimetric sets.

math.DG↗