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Kai Rajala

Publications and source records attributed to Kai Rajala.

At least 19 recordsLinked to original sources
arXiv2026

Length distortion of volume-preserving Lipschitz mappings

We show that every area-preserving Lipschitz map between metric surfaces distorts the length of almost every curve by at most a multiplicative factor, answering a question posed by the first author and Ntalampekos. We give two different proofs. The first is entirely intrinsic and extends to higher dimensions under additional assumptions. The second relies on non-smooth uniformization theory and yields the conclusion that the family of curves intersecting the purely 2-unrectifiable part of a metric surface in a set of positive length is exceptional.

math.MG↗
arXiv2025

Exhaustions of circle domains

Koebe's conjecture asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. We prove that every domain $Ω$ satisfying Koebe's conjecture admits an exhaustion, i.e., a sequence of interior approximations by finitely connected domains, so that the associated conformal maps onto finitely connected circle domains converge to a conformal map $f$ from $Ω$ onto a circle domain. Thus, if Koebe's conjecture is true, it can be proved by utilizing interior approximations of a domain. The main ingredient in the proof is the construction of quasiround exhaustions of a given circle domain $Ω$. In the case of such exhaustions, if $\partial Ω$ has area zero, we show that $f$ is a Möbius transformation. The paper builds upon a range of tools, including planar topology, Voronoi cells, classical and modern methods in (quasi)conformal mapping theory, the transboundary modulus of Schramm, and the dynamics of Schottky groups.

math.CV↗
arXiv2025

Uniformization of cofat domains on metric two-spheres

We extend \emph{Schramm's cofat uniformization theorem} to cofat domains on upper Ahlfors 2-regular metric two-spheres $X$. Specifically, we show that if $\Omega \subset X$ is a cofat domain, then there exists a $\frac{\pi}{2}$-quasiconformal homeomorphism $f: \Omega \to D$ onto a circle domain $D \subset \mathbb{S}^2$. Moreover, $f$ preserves the point-components and non-trivial complementary components. We also construct examples which show that the above conclusions are not true for countably connected $\ell^{\alpha}$-subdomains of $\mathbb{S}^2$.

math.CV↗
arXiv2025

Conformal Uniformization of Domains Bounded by Quasitripods

We prove Koebe's conjecture and a version of Schramm's cofat uniformization theorem for domains $Ω\subset \mathbb C$ satisfying conditions involving quasitripods, i.e., quasisymmetric images of the standard tripod. If the non-point complementary components of $Ω$ contain uniform quasitripods with large diameters and satisfy a packing condition, then there exists a conformal map $f\colonΩ\to D$ onto a circle domain $D$. Moreover, $f$ preserves the classes of point-components and non-point components. The packing condition is satisfied if $Ω$ is cospread, i.e., if the complementary components contain uniform quasitripods in all scales.

math.CV↗
arXiv2024

Rigid circle domains with non-removable boundaries

We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain $Ω$ on the Riemann sphere with conformally non-removable boundary. Here rigidity means that every conformal map from $Ω$ onto another circle domain is a Möbius transformation, and non-removability means that there is a homeomorphism of the Riemann sphere which is conformal off $\partial Ω$ but not everywhere. Our construction is based on a theorem of Wu, which states that the product of any Cantor set $E$ with a sufficiently thick Cantor set $F$ is non-removable. We show that one can choose $E$ and $F$ so that the complement of the union of $E \times F$ and suitably placed disks is rigid. The proof of rigidity involves a metric characterization of conformal maps, which was recently proved by Ntalampekos. The other direction of the rigidity conjecture, i.e., whether removability of the boundary implies rigidity, remains open.

math.CV↗
arXiv2024

Extremal length and duality

Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In ``Extremal length and functional completion'', Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings: Extremal length and uniformization of metric surfaces, Extremal length of families of surfaces and quasiconformal maps between $n$-dimensional spaces, and Schramm's transboundary extremal length and conformal maps between multiply connected plane domains.

math.CV↗
arXiv2024

Mappings of finite distortion on metric surfaces

We investigate basic properties of mappings of finite distortion $f:X \to \mathbb{R}^2$, where $X$ is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite $2$-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant $f:X \to \mathbb{R}^2$ with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if $f$ is moreover injective then $f^{-1}$ is a Sobolev map.

math.MG↗
arXiv2024

Definitions of quasiconformality on metric surfaces

We explore the interplay between different definitions of distortion for mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite 2-dimensional Hausdorff measure. We establish that finite distortion in terms of the familiar analytic definition always implies finite distortion in terms of maximal and minimal stretchings along paths. The converse holds for maps with locally integrable distortion. In particular, we prove the equivalence of various notions of quasiconformality, implying a novel uniformization result for metric surfaces.

math.MG↗
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 κ \int_{ X } g ρ \,d\mathcal{H}^{2} \quad\text{for every Borel $g \colon X \rightarrow \left[0,\infty\right]$,} \end{equation*} where $ρ$ is any integrable upper gradient of $u$. If $ρ$ is locally $L^2$-integrable, we obtain the sharp constant $κ=4/π$. 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

Uniformization of planar domains by exhaustion

We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain $D \subset \hat{C}$ admits exhaustions, i.e., increasing sequences of finitely connected subdomains $D_j$ whose union is $D$. By Koebe's theorem, each $D_j$ admits a conformal map $f_{D_j}$ from $D_j$ onto a circle domain $f_{D_j}(D_j)$. Assuming $f_{D_j} \to f$, our goal is to find out if $f(D)$ is also a circle domain. We present a countably connected $D$ with an exhaustion $(D_j)$ so that $(f_{D_j})$ has a limit whose image is not a circle domain, and a domain $Ω$ with an exhaustion $(Ω_j)$ so that $(f_{Ω_j})$ has a limit whose image has uncountably many non-point complementary components. On the other hand, we prove that every exhaustion $(D_j)$ of a countably connected $D$ admits a refinement so that the image of the corresponding limit map is a circle domain. Our result extends the He-Schramm theorem on the uniformization of countably connected domains and provides a new proof.

math.CV↗
arXiv2021

Uniformization with infinitesimally metric measures

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $μ$ on such a space, we introduce $μ$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $μ$. We show that if $μ$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $μ$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

math.CV↗
arXiv2019

Duality of moduli in regular metric spaces

F. Gehring and W. Ziemer proved that the p-modulus of the family of paths connecting two continua is dual to the p^*-modulus of the corresponding family of separating hypersurfaces. In this paper we show that a similar result holds in complete Ahlfors-regular metric spaces that support a weak 1-Poincaré inequality. As an application we obtain a new characterization for quasiconformal mappings between such spaces.

math.MG↗
arXiv2019

Reciprocal lower bound on modulus of curve families in metric surfaces

We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $ζ_1, ζ_2, ζ_3, ζ_4$ and let $Γ(ζ_i, ζ_j; Q)$ denote the family of curves in $Q$ connecting $ζ_i$ and $ζ_j$; then $\text{mod} Γ(ζ_1, ζ_3; Q) \text{mod} Γ(ζ_2, ζ_4; Q) \geq 1/κ$ for $κ= 2000^2\cdot (4/π)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in $\mathbb{R}^2$.

math.MG↗
arXiv2017

Quasispheres and metric doubling measures

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking.

math.CV↗
arXiv2016

Uniformization of two-dimensional metric surfaces

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by quasisymmetric maps.

math.CV↗
arXiv2014

Wolfe's theorem for weakly differentiable cochains

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension $m$ in $\mathbb{R}^n$ with the space of flat $m$-cochains, that is, the dual space of flat chains of dimension $m$ in $\mathbb{R}^n$. The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in $\mathbb{R}^n$. A suitable theory of Sobolev cochains has recently been initiated by the second and third author. It is based on the concept of upper norm and upper gradient of a cochain, introduced in analogy with Heinonen-Koskela's concept of upper gradient of a function.

math.AP↗