Mario Bonk, Bruce Kleiner
We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standard 2-sphere, answering a question of Heinonen and Semmes.
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We consider decomposition spaces $\R^3/G$ that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on $\R^3/G$ constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space $\R^3/G\times \R^m$ imposes quantitative topological constraints, in terms of the circulation and growth, to the defining sequences for $\R^3/G$. We give a necessary condition and a sufficient condition for the existence of parametrization. The necessary condition answers a qu…
Let $Ω$ be a planar Jordan domain and $α>0$. We consider double-dome-like surfaces $Σ(Ω,t^α)$ over $\overlineΩ$ where the height of the surface over any point $x\in\overlineΩ$ equals $\text{dist}(x,\partialΩ)^α$. We identify the necessary and sufficient conditions in terms of $Ω$ and $α$ so that these surfaces are quasisymmetric to $\mathbb{S}^2$ and we show that $Σ(Ω,t^α)$ is quasisymmetric to the unit sphere $\mathbb{S}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.
We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is quasisymmetrically equivalent to a model surface of the same topology. Moreover, we show that this is also true for surfaces as above with non-empty boundary and that the corresponding map can be chosen in a canonical way. Our proof is based on a local argument involving the existence of quasisymmetric parametrizations for metric discs as shown in in a paper of Lytchak and Wenger.
Quasisymmetric rigidity of the Brownian sphere
本文证明了布朗球体（布朗映射）几乎必然没有非平凡的拟对称自同构，且两个独立的布朗球体几乎必然不是拟对称等价的，从而建立了其拟对称刚性。
Bonk and Kleiner showed that any metric sphere which is Ahlfors 2-regular and linearly locally contractible is quasisymmetrically equivalent to the standard sphere, in a quantitative way. We extend this result to arbitrary metric compact orientable surfaces.
Let $Ω$ be a planar Jordan domain. We consider double-dome-like surfaces $Σ$ defined by graphs of functions of $dist( \cdot ,\partial Ω)$ over $Ω$. The goal is to find the right conditions on the geometry of the base $Ω$ and the growth of the height so that $Σ$ is a quasisphere, or quasisymmetric to $\mathbb{S}^2$. An internal uniform chord-arc condition on the constant distance sets to $\partial Ω$, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in $\mathbb{R}^n$, for any $n\ge 3$.
We look for minimal conditions on a two-dimensional metric surface $X$ of locally finite Hausdorff $2$-measure under which $X$ admits an (almost) parametrization with good geometric and analytic properties. Only assuming that $X$ is locally geodesic, we show that Jordan domains in $X$ of finite boundary length admit a quasiconformal almost parametrization. If $X$ satisfies some further conditions then such an almost parametrization can be upgraded to a geometrically quasiconformal homeomorphism or a quasisymmetric homeomorphism. In particular, we recover Rajala's recent quasiconformal uniformi…
We show that for all $n \geq 2$, there exists a doubling linearly locally contractible metric space $X$ that is topologically a $n$-sphere such that every weak tangent is isometric to $\R^n$ but $X$ is not quasisymmetrically equivalent to the standard $n$-sphere. The same example shows that $2$-Ahlfors regularity in Theorem 1.1 of \cite{BK02} on quasisymmetric uniformization of metric $2$-spheres is optimal.
We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a Transboundary Loewner Property (TLP) which is a transboundary analogue of the Loewner property of Heinonen and Koskela. We show that a dyadic slit carpet can be quasisymmetrically embedded into the plane if and only if it is TLP. Moreover, every dyadic slit carpet $X$ can be associated to a "pillowcase sphere" $\widehat{X}$ which is a metric space homeomorphic…
The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal structure. We give a similar uniformization for Ahlfors 2-regular, linearly locally connected metric planes; instead of conformal equivalence, we are concerned with quasisymmetric equivalence.
We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected domain in the standard 2-sphere is quasisymmetrically equivalent to a circle domain if and only if X is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.
We use the recently established existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parametrizations of linearly locally connected, Ahlfors $2$-regular metric $2$-spheres. Generalizations and applications to the geometry of such surfaces are described.
In this paper, we study the quasisymmetric embeddability of weak tangents of metric spaces. We first show that quasisymmetric embeddability is hereditary, i.e., if $X$ can be quasisymmetrically embedded into $Y$, then every weak tangent of $X$ can be quasisymmetrically embedded into some weak tangent of $Y$, given that $X$ is proper and doubling. However, the converse is not true in general; we will illustrate this with several counterexamples. In special situations, we are able to show that the embeddability of weak tangents implies global or local embeddability of the ambient space. Finally,…
The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, more abstract, metric spaces. Similarly, Liouville Quantum Gravity has been shown to be "equivalent" to the Brownian map but the exact nature of the correspondence (i.e.\ embedding) is still unknown. In this article we show that any embedding of the Brownian map or continuum random tree…
A doubling Loewner sphere that is not a quasisphere
构造了一个满足加倍、线性局部连通和Loewner性质但非拟球面的度量2-球面，否定了Ntalampekos的相关问题。
The conformal dimension of the Brownian sphere is two
研究了布朗树的共形维度，证明其等于拓扑维度2，处于2到4之间。
The metric geometry of paper surfaces under geometric constraints
本文研究在几何约束下纸面的度量几何，证明其满足Ahlfors 2-正则性和线性局部可缩性，从而保证存在到标准2球面的准等距参数化。
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 give…
Pullback metrics and homological obstructions to BLD remetrization of branched covers
本文刻画球面分支自覆盖的 BLD 重新度量化，通过同调障碍证明多孔性必要但不充分，并给出二维情形的拟对称不变性。
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 infinite…
Uniformization of metric surfaces: A survey
本文综述了度量曲面均匀化的最新进展，从经典共形均匀化到分形曲面上的拟对称与拟共形均匀化，最终介绍任意局部有限面积度量曲面的弱拟共形均匀化结果。
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 quasiconforma…
We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4π)^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak--Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau's problem in a CAT($0$) space have only isolated branch points. We also show that any …
We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala-Wenger. These results have been previously established by the authors under the assumption that the metric surface…
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel-Niels…
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