arXiv subjects

Toni Ikonen

Publications and source records attributed to Toni Ikonen.

At least 19 recordsLinked to original sources

arXiv2026

Systolic inequalities for metric surfaces via filling minimality

We prove optimal systolic inequalities for length spaces homeomorphic to a torus of genus one or a real projective plane. In both cases, the optimal constant coincides with the constant from the (reversible) Finsler setting. This generalizes the classical results for Riemannian and Finsler surfaces. The proof of the inequality for the torus relies on an analysis of the asymptotic volume growth of the universal cover, together with a strengthening of the previously known area minimality of two-dimensional normed planes. For the inequality of the real projective plane, we similarly extend a minimality result of hemispheres. These results build upon works of Burago-Ivanov and Ivanov, respectively. In their proofs we apply recent uniformization theorems for metric disks and the theory of area-minimizing disks in metric spaces due to Lytchak-Wenger.

math.MG↗

arXiv2026

A rescaling principle for quasiregular curves with applications to hyperbolicity

We prove a Miniowitz--Zalcman rescaling principle for quasiregular curves into calibrated manifolds. We have two main applications. First, we introduce Brody hyperbolicity adapted to our setting and prove its equivalence to the normality of the family of quasiregular curves from the Euclidean unit ball into the target. When normality holds, we quantify the local modulus of continuity for quasiregular curves using an injectivity radius lower bound and a sectional curvature upper bound of the target. Second, in the special case of conformal curves into closed calibrated manifolds, we prove the equivalence of Kobayashi and Brody hyperbolicity. This answers a question posed by Broder--Iliashenko--Madnick. As an intermediate result, we prove an analogue of Marty's theorem from complex analysis in this setting. Additionally, we construct new examples of non-constant entire quasiregular curves factoring through a Special Lagrangian submanifold of a closed Calabi--Yau manifold, an associative submanifold of a closed $G_2$ manifold, and four-dimensional analogues thereof, providing obstructions to Brody hyperbolicity.

math.DG↗

arXiv2026

Sobolev-to-Lipschitz property of geodesically complete spaces with curvature bounded from above

We prove that every length space with curvature bounded from above that is geodesically complete has the Sobolev-to-Lipschitz property with exponent infinity. That is, every Sobolev map in the $W^{1,\infty}$-space has a Lipschitz representative so that the Lipschitz constant coincides with the infinity energy of the map. The proof is geometric and relies on arbitrarily small perturbations of geodesics to a curve that has zero length on the singular set. The motivation is to develop the analytic theory of such spaces; in particular, our result implies that GCBA spaces satisfy the infinity Poincar\'e inequality and an essential assumption in the theory of Lipschitz-Volume rigidity.

math.DG↗

arXiv2026

Ultralimits of Sobolev maps and stability of Dehn functions

We show that the ultralimit of a bounded sequence of Lipschitz maps into pointed metric spaces extends naturally to $p$-bounded sequences of Sobolev maps and that this ultralimit for Sobolev maps enjoys desirable properties. We use this to prove the stability of Dehn functions under ultraconvergence of pointed length spaces, thus resolving a problem posed by several researchers in the field. As an application, we obtain a simpler proof of a recent result of Stadler--Wenger, previously proved in the locally compact case by Lytchak--Wenger, characterizing spaces of curvature bounded above by $\kappa$ via an isoperimetric inequality for curves.

math.DG↗

arXiv2025

Metric Sobolev spaces II: dual energies and divergence measures

This is the second of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several approaches to vector calculus in the non-smooth setting of complete and separable metric spaces equipped with a boundedly-finite Borel measure. More precisely, we study different notions of (co)vector fields and derivations appearing in the literature, as well as their mutual relation. We also carry forward a thorough investigation of gradients, divergence measures, and Laplacian measures, together with their applications in potential analysis (for example, regarding the condenser capacity) and in the study of duality properties of Sobolev spaces. Most of the results are obtained for the full range of exponents $p\in[1,\infty)$ and without finiteness assumption on the measure.

math.FA↗

arXiv2025

Entire conformal curves are affine or have super-Euclidean energy growth

We prove that entire conformal curves $\mathbb{R}^n \rightarrow \mathbb{R}^m$ fall into two classes: either the curve is affine or the average energy in a ball is strictly increasing for large radii and diverges to infinity. This rigidity follows from a blow-down argument and the strong interaction of the generalized Cauchy--Riemann equations with calibrated geometries and the sharp isoperimetric inequality for integral currents due to Almgren. As an application, we prove that every entire Lipschitz conformal curve is affine. This can be considered a higher-dimensional analog of the statement that an entire holomorphic function with a bounded complex differential is affine. We also recall that a certain punctured cone over a Legendrian torus provides a submanifold in $\mathbb{R}^n$, for $n \geq 3$, for which an entire conformal curve yields a conformal covering map. As a related result, we prove that if a non-constant entire conformal curve factors through an $n$-dimensional submanifold, then the submanifold is calibrated and conformally equivalent to a flat non-compact quotient of $\mathbb{R}^n$.

math.DG↗

arXiv2025

On infinity thick quasiconvexity and applications

We investigate geometric properties of a metric measure space where every function in the Newton--Sobolev space $N^{1,\infty}(Z)$ has a Lipschitz representative. We prove that when the metric space is locally complete and the reference measure is infinitesimally doubling, the above property is equivalent to the space being very $\infty$-thick quasiconvex up to a scale. That is, up to some scale, every pair of points can be joined by a family of quasiconvex curves that is not negligible for the $\infty$-modulus. As a first application, we prove a local-to-global improvement for the weak $(1,\infty)$-Poincar\'e inequality for locally complete quasiconvex metric spaces that have a doubling reference measure. As a second application, we apply our results to the existence and uniqueness of $\infty$-harmonic extensions with Lipschitz boundary data for precompact domains in a large class of metric measure spaces. As a final application, we illustrate that in the context of Sobolev extension sets, very $\infty$-thick quasiconvexity up to a scale plays an analogous role as local uniform quasiconvexity does in the Euclidean space. Our assumptions are adapted to the analysis of Sobolev extension sets and thus avoid stronger assumptions such as the doubling property of the measure. Examples satisfying our assumptions naturally occur as simplicial complexes, GCBA spaces, and metric quotients of Euclidean spaces.

math.MG↗

arXiv2025

Smooth contact lifts to central extensions of Carnot groups

We consider the existence problem of lifting a smooth contact map between Carnot groups to a smooth contact map between central extensions of the original groups. Our main result is a necessary and sufficient criterion formulated using the pullback of any de Rham potential of the codomain central extension 2-cocycle: the Rumin differential of the pullback is in a linear image of the domain central extension 2-cocycle. We also show a necessary criterion using the Pansu pullback: the Pansu pullback of the codomain central extension 2-cocycle and a linear image of the domain central extension 2-cocycle are in the same Lie algebra cohomology class. We prove that the latter criterion is sufficient if the domain group is Lipschitz 1-connected, or if the pullback has maximal weight among Lie algebra 2-cohomology classes.

math.DG↗

arXiv2024

Liouville's theorem in calibrated geometries

We consider the following extension of the classical Liouville theorem: A calibration $\omega \in \Lambda^n \mathbb{R}^m$, where $3 \le n \le m$, has the Liouville property if a Sobolev mapping $F\colon \Omega \to \mathbb{R}^m$, where $\Omega \subset \mathbb{R}^n$ is a domain, in $W^{1,n}_{loc}( \Omega, \mathbb{R}^m )$ satisfying $\|DF\|^n = \star F^{*}\omega$ almost everywhere is a restriction of a M\"obius transformation $\mathbb{S}^m \to \mathbb{S}^m$. We show that, for $m\ge 5$, every calibration in $\Lambda^{m-2} \mathbb{R}^m$ has the Liouville property and, in low dimensions, a calibration $\omega \in \Lambda^n \mathbb{R}^m$ has the Liouville property for $3 \le n \le m \le 6$ unless $\omega$ is face equivalent to the Special Lagrangian. In these cases, the Liouville property stems from isoperimetric rigidity of these mappings together with a classification of calibrations whose conformally flat calibrated submanifolds are flat. We also show that, for $3 \leq n \leq m$, the calibrations with the Liouville property form a dense $G_\delta$ set in the space of calibrations. As an application, we consider factorization of more general quasiregular curves and stability of quasiregular curves of small distortion.

math.DG↗

arXiv2024

Quasiregular curves: Removability of singularities

We prove a Painlev\'e theorem for bounded quasiregular curves in Euclidean spaces extending removability results for quasiregular mappings due to Iwaniec and Martin. The theorem is proved by extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves. We also establish a qualitatively sharp removability theorem for quasiregular curves whose target is a Riemannian manifold with sectional curvature bounded from above and an injectivity radius lower bound. As an application, we extend a theorem of Bonk and Heinonen for quasiregular mappings to the setting of quasiregular curves: every non-constant quasiregular $\omega$-curve from $\mathbb{R}^n$ into $( N, \omega )$, where the bounded cohomology class of $\omega$ is in the bounded K\"unneth ideal, has infinite energy.

math.DG↗

arXiv2024

Metric Sobolev spaces I: equivalence of definitions

This is the first of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several notions of metric Sobolev space and on their equivalence. More precisely, we give a systematic presentation of first-order $p$-Sobolev spaces, with $p\in[1,\infty)$, defined over a complete and separable metric space equipped with a boundedly-finite Borel measure. We focus on three different approaches: via approximation with Lipschitz functions; by studying the behaviour along curves, in terms either of the curve modulus or of test plans; via integration-by-parts, using Lipschitz derivations with divergence. Eventually, we show that all these approaches are fully equivalent. We emphasise that no doubling or Poincar\'{e} assumption is made, and that we allow also for the exponent $p=1$. A substantial part of this work consists of a self-contained and partially-revisited exposition of known results, which are scattered across the existing literature, but it contains also several new results, mostly concerning the equivalence of metric Sobolev spaces for $p=1$.

math.FA↗

arXiv2023

Pushforward of currents under Sobolev maps

We prove that a Sobolev map from a Riemannian manifold into a complete metric space pushes forward almost every compactly supported integral current to an Ambrosio--Kirchheim integral current in the metric target, where "almost every" is understood in a modulus sense. As an application, we prove that when the target supports an isoperimetric inequality of Euclidean type for integral currents, an isoperimetric inequality for Sobolev mappings relative to bounded, closed and additive cochains follows. Using the results above, we answer positively to an open question by Onninen and Pankka on sharp H\"older continuity for quasiregular curves. A key tool in the continuity proof is Almgren's isoperimetric inequality for integral currents.

math.DG↗

arXiv2022

Extensions and approximations of Banach-valued Sobolev functions

In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincar\'e inequality, we investigate when a given Banach-valued Sobolev function defined on a subset satisfying a measure-density condition is the restriction of a Banach-valued Sobolev function defined on the whole space. We investigate the problem for Haj{\l}asz- and Newton-Sobolev spaces, respectively. First, we show that Haj{\l}asz-Sobolev extendability is independent of the target Banach spaces. We also show that every $c_0$-valued Newton-Sobolev extension set is a Banach-valued Newton-Sobolev extension set for every Banach space. We also prove that any measurable set satisfying a measure-density condition and a weak Poincar\'e inequality up to some scale is a Banach-valued Newton-Sobolev extension set for every Banach space. Conversely, we verify a folklore result stating that when $n\leq p<\infty$, every $W^{1,p}$-extension domain $\Omega \subset \mathbb{R}^n$ supports a weak $(1,p)$-Poincar\'e inequality up to some scale. As a related result of independent interest, we prove that in any metric measure space when $1 \leq p < \infty$ and real-valued Lipschitz functions with bounded support are norm-dense in the real-valued $W^{1,p}$-space, then Banach-valued Lipschitz functions with bounded support are energy-dense in every Banach-valued $W^{1,p}$-space whenever the Banach space has the so-called metric approximation property.

math.FA↗

arXiv2022

Coarea Inequality for Monotone Functions on Metric Surfaces

We study coarea inequalities for metric surfaces -- metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure $\mathcal{H}^2$. For monotone Sobolev functions $u\colon X \to \mathbb{R} $, we prove the inequality \begin{equation*} \int_{ \mathbb{R} }^{*} \int_{ u^{-1}(t) } g \,d\mathcal{H}^{1} \,dt \leq \kappa \int_{ X } g \rho \,d\mathcal{H}^{2} \quad\text{for every Borel $g \colon X \rightarrow \left[0,\infty\right]$,} \end{equation*} where $\rho$ is any integrable upper gradient of $u$. If $\rho$ is locally $L^2$-integrable, we obtain the sharp constant $\kappa=4/\pi$. The monotonicity condition cannot be removed as we give an example of a metric surface $X$ and a Lipschitz function $u \colon X \to \mathbb{R}$ for which the coarea inequality above fails.

math.MG↗

arXiv2021

Pullback of a quasiconformal map between arbitrary metric measure spaces

We prove that every (geometrically) quasiconformal homeomorphism between metric measure spaces induces an isomorphism between the cotangent modules constructed by Gigli. We obtain this by first showing that every continuous mapping $\varphi$ with bounded outer dilatation induces a pullback map $\varphi^*$ between the cotangent modules of Gigli, and then proving the functorial nature of the resulting pullback operator. Such pullback is consistent with the differential for metric-valued locally Sobolev maps introduced by Gigli-Pasqualetto-Soultanis. Using the consistency between Gigli's and Cheeger's cotangent modules for PI spaces, we prove that quasiconformal homeomorphisms between PI spaces preserve the dimension of Cheeger charts, thereby generalizing earlier work by Heinonen-Koskela-Shanmugalingam-Tyson. Finally, we show that if $\varphi$ is a given homeomorphism with bounded outer dilatation, then $\varphi^{-1}$ has bounded outer dilatation if and only if $\varphi^{*}$ is invertible and $\varphi^{-1}$ is Sobolev. In contrast to the setting of Euclidean spaces, Carnot groups, or more generally, Ahlfors regular PI spaces, the Sobolev regularity of $\varphi^{-1}$ needs to be assumed separately.

math.MG↗

arXiv2021

Two-dimensional metric spheres from gluing hemispheres

We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving homeomorphism mapping the equator onto itself, where the distance on Z is the canonical distance that is locally isometric to the spherical distance off the seam. We show that if Z is quasiconformally equivalent to the sphere, in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves. We also show that g is bi-Lipschitz if and only if Z has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping from the Euclidean sphere onto itself. Furthermore, we show that if the inverse of g is absolutely continuous and g admits a homeomorphic extension with exponentially integrable distortion, then Z is quasiconformally equivalent to the Euclidean sphere.

math.CV↗

arXiv2020

Abstract and concrete tangent modules on Lipschitz differentiability spaces

We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from a recent article by Bate--Kangasniemi--Orponen, this equivalence is used to show that the ${\rm Lip}-{\rm lip}$ -type condition ${\rm lip} f\le C|Df|$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to ${\rm lip} f =|Df|$. We also provide a direct proof of a result by Gigli and the second author that, for a space with a strongly rectifiable decomposition, Gigli's tangent module admits an isometric embedding into the so-called Gromov--Hausdorff tangent module, without any a priori reflexivity assumptions.

math.MG↗

arXiv2020

Quasiconformal Jordan domains

We extend the classical Carath\'eodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has finite Hausdorff $2$-measure, the boundary $\partial Y = \overline{Y} \setminus Y$ is homeomorphic to $\mathbb{S}^{1}$, and there exists a homeomorphism $\phi \colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $\phi$ has a continuous, monotone, and surjective extension $\Phi \colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for $\Phi$ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of $\Phi$ to $\mathbb{S}^{1}$ being a quasisymmetry and to $\partial Y$ being bi-Lipschitz equivalent to a quasicircle in the plane.

math.MG↗