Christopher J. Bishop
We give an example of a totally disconnected set E in R^3 which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism f of R^3 to itself which is quasiconformal off E, but not quasiconformal on all of R^3. The set E may be taken with Hausdorff dimension 2. The construction also gives a non-removable set for locally biLipschitz homeomorphisms.
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We establish sharp bounds for simultaneous local rotation and Hölder-distortion of planar quasiconformal maps. In addition, we give sharp estimates for the corresponding joint quasiconformal multifractal spectrum, based on new estimates for Burkholder functionals with complex parameters. As a consequence, we obtain optimal rotation estimates also for bi-Lipschitz maps.
Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2 and that $D\varsubsetneq E$ and $D'\varsubsetneq E'$ are uniform domains with homogeneously dense boundaries. We consider the class of all $φ$-FQC (freely $φ$-quasiconformal) maps of $D$ onto $D'$ with bilipschitz boundary values. We show that the maps of this class are $η$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class $φ$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz b…
We study integral curvature conditions for a Riemannian metric $g$ on $S^4$ that quantify the best bilipschitz constant between $(S^4,g)$ and the standard metric on $S^4$. Our results show that the best bilipschitz constant is controlled by the $L^2$-norm of the Weyl tensor and the $L^1$-norm of the $Q$-curvature, under the conditions that those quantities are sufficiently small, $g$ has a positive Yamabe constant and the $Q$-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a posi…
In this paper we prove that if $ϕ:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq c^{-1} (\frac{γ(ϕ(E))}{\diam(ϕ(B))})^{\frac{2K}{K+1}},$$ where $γ$ stands for the analytic capacity and $\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ is a capacity associated to a non linear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e. removable for bounded $K$-quasiregular maps), it has positive capacity $\dot C_{frac{2K}{2K+1},\frac{2K+1}{K+1…
We study quasiconformal maps on non-rigid Carnot groups equipped with Carnot metric. We show that for most non-rigid Carnot groups N, all quasiconformal maps on N must be biLipschitz.
We show that quasiconformal maps on many Carnot groups must be biLipschitz. In particular, this is the case for 2-step Carnot groups with reducible first layer. These results have implications for the rigidity of quasiisometries between negatively curved solvable Lie groups.
Metric surfaces and conformally removable sets in the plane
利用度量曲面理论，刻画了平面中共形可移除集，证明了S-可移除与CH-可移除的充要条件，并指出映射到度量曲面而非平面时命题成立。
The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, $φ$-uniformity and $δ$-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of $\mathbb{R}^n$ are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that $φ$-uniformit…
We study homeomorphisms of controlled $p$-module by certain integrals. In this way, we establish various properties of mappings and show that their features are close to quasiconformal and bilipschitz mappings.
CNED sets: countably negligible for extremal distances
本文证明多类已知拟共形可移除集均为 CNED，并给出闭集为 (C)NED 的充要条件及可数并封闭性，从而统一并扩展可移除性理论。
We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop. The proof involves a new technique of constructing an exceptional homeomorphism from $\mathbb R^2$ into some non-planar surface $S$, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk-Kleiner Theorem arXiv:math/0107171. We also prove that all homeomorphic copies of the Sierpiński gasket are non-removable for continuous Sobolev functions of the class $W^{1,p}$ for $1\leq p\leq 2$, thus complementing and sharpening the results of the …
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…
Let $α\in(0,1)$, $K\geq 1$, and $d=2\frac{1+αK}{1+K}$. Given a compact set $E\subset\C$, it is known that if $\H^d(E)=0$ then $E$ is removable for $α$-Hölder continuous $K$-quasiregular mappings in the plane. The sharpness of the index $d$ is shown with the construction, for any $t>d$, of a set $E$ of Hausdorff dimension $\dim(E)=t$ which is not removable. In this paper, we improve this result and construct compact nonremovable sets $E$ such that $0<\H^d(E)<\infty$. For the proof, we give a precise planar $K$-quasiconformal mapping whose Hölder exponent is strictly bigger than $\frac{1}{K}$, a…
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs {enumerate} [$\bullet$] Given $α$ and $β$ in $[0,2]$, there exists a David map $ϕ:\CC \to \CC$ and a compact set $Λ$ such that $\Hdim Λ=α$ and $\Hdim ϕ(Λ)=β$. \vs [$\bullet$] There exists a David map $ϕ:\CC \to \CC$ such that the Jordan curve $Γ=ϕ(\Sen)$ satisfies $\Hdim Γ=2$.\vs {enumerate} One should contrast the first statement with the fact that quasiconformal m…
The classical Painlevé theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $σ$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not enough to guarantee this property. We also study absolute continuity pro…
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