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Zero-length compact sets and quasicircles are conformally removable

Statement

Assume the Axiom of Choice. Let $H_{χ}^{1}$ denote one-dimensional Hausdorff measure for the chordal metric on $\hat{C}$ . Every compact set $K \subseteq \hat{C}$ with $H_{χ}^{1} (K) = 0$ is globally conformally removable; the same proof shows this for every compact $K$ with $H_{χ}^{1} (K) < \infty$ (Conformal removability of compact sets).

Every quasicircle is globally conformally removable (Quasicircles, quasidisks, quasiarcs, and quasilines).

No converse and no Hausdorff-dimension threshold are asserted.

Facts & Assumptions

Given: AC and a compact set $K \subseteq \hat{C}$ with finite chordal one-dimensional Hausdorff measure.

[F1]

The chordal metric is Euclidean distance after stereographic projection. The finite-coordinate formula $χ (z, w) = \frac{2 ∣ z - w ∣}{\sqrt{(1 + ∣ z ∣^{2}) (1 + ∣ w ∣^{2})}}$ follows by expanding the squared distance between the coordinate images in Stereographic projection identifies the Riemann sphere with the unit two-sphere; it gives bi-Lipschitz equivalence to Euclidean distance on bounded chart disks (The chordal metric on the Riemann sphere). Hausdorff measure is defined by small-diameter covers; planar Lebesgue outer measure is countably subadditive and a square has its positive Euclidean area (Unnormalised Hausdorff measure, Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ , A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F2]

Hausdorff measure is monotone and is multiplied by at most $L$ under an $L$ -Lipschitz map; the latter follows directly by mapping the covers in Unnormalised Hausdorff measure. Every Möbius map is chordally Lipschitz: if $M (z) = (a z + b) / (c z + d)$ has coefficient matrix $A$ , then $χ (M z, M w) = \frac{2 ∣ \det A ∣ ∣ z - w ∣}{∥ A (z, 1) ∥_{2} ∥ A (w, 1) ∥_{2}},$ with the formula extended continuously at poles and infinity. If $s_{\min} > 0$ is the smallest singular value of $A$ , comparison with [F1] gives $χ (M z, M w) \leq ∣ \det A ∣ s_{\min}^{} χ (z, w)$ .

[F3]

Möbius transformations are biholomorphic in the sphere charts and form a group under composition (Every Möbius transformation is a biholomorphism of the Riemann sphere, Möbius transformations form a group and identify with the projective linear quotient of GL_2(C), The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity).

[F4]

If a compact $E \subseteq \hat{C}$ has finite chordal $H_{χ}^{1}$ and $u : \hat{C} \to C$ is continuous and holomorphic off $E$ , then $u$ is constant (Compact sets of finite length are removable for continuous analytic functions). This supplier states the required finite-length result. Its current local proof uses a finite overlapping Hausdorff-square cover, polygon-cell cancellation and continuity, supplying the missing covering/contour argument independently of Garnett.

[F5]

Global conformal removability means that every sphere homeomorphism conformal off the compact set is Möbius (Conformal removability of compact sets). That definition proves the neighborhood-local equivalence under AC using its explicit MRMT/extension route; this theorem uses only the choice-free global predicate, so the equivalence is not an input here.

[F6]

The round circle $S^{1}$ is globally conformally removable (Round circles and straight lines are conformally removable). Its round-circle gluing proof and the earlier one-quasiconformal criterion supply the assertion.

[F7]

Global conformal removability is invariant under quasiconformal sphere homeomorphisms (Conformal removability is invariant under quasiconformal maps). Its coefficient-straightening proof consumes the stable13 sphere MRMT and the earlier area/inverse-N interfaces.

[F8]

By definition, a quasicircle is the image of $S^{1}$ under a quasiconformal sphere homeomorphism (Quasicircles, quasidisks, quasiarcs, and quasilines). The earlier analytic/geometric equivalence supplies those conventions.

[F9]

AC implies Countable Choice (AC implies DC implies countable choice).

[F10]

An injective holomorphic map on a complex domain has nonzero derivative (An injective holomorphic map has no critical point and is biholomorphic onto its image).

Proof

technique · normalize the exceptional set away from infinity, form a continuous holomorphic quotient, and apply finite-length removability; transport round-circle removability to quasicircles by quasiconformal invariance

1.1F1F9algebra

Every nonempty open subset of the sphere has infinite chordal $H_{χ}^{1}$ . Indeed, it contains a closed Euclidean square in a finite chart, and [F1] compares the two metrics there. If sets of Euclidean diameters $d_{j} \leq δ$ cover that square, each has planar outer area at most $π d_{j}^{2} \leq π δ d_{j}$ ; countable subadditivity gives $∣ Q ∣ \leq π δ \sum_{j} d_{j}$ , so the covering sums tend to infinity as $δ ↓ 0$ . Thus $K$ has empty interior and in particular is not the whole sphere.

2.1F2F3step 1.1construct

Choose $p \in \hat{C} ∖ K$ , and let $T$ be the identity if $p = \infty$ and $T (z) = 1 / (z - p)$ otherwise. Then $T$ is Möbius, $K^{'} : = T (K)$ is compact in $C$ , and [F2] gives $H_{χ}^{1} (K^{'}) < \infty$ .

3.1F3F10step 2.1algebra

Let $F$ be any sphere homeomorphism conformal off $K$ . Define $S$ to be the identity if $(T \circ F \circ T^{- 1}) (\infty) = \infty$ and otherwise set $S (z) = 1 / (z - (T \circ F \circ T^{- 1}) (\infty))$ . The map $F_{0} : = S \circ T \circ F \circ T^{- 1}$ fixes infinity and is conformal off $K^{'}$ . Since $K^{'}$ is compact in $C$ , $F_{0}$ is conformal near infinity. In the local coordinate $w = 1 / z$ , the chart expression $H (w) = 1 / F_{0} (1 / w)$ is holomorphic, injective, and vanishes at $0$ . By [F10], $H^{'} (0) = c \neq 0$ ; its Taylor expansion $H (w) = c w + d w^{2} + O (w^{3})$ therefore yields $F_{0} (z) = a z + b + O (1 / z)$ near infinity, with $a = c^{- 1} \neq 0$ .

4.1step 2.1step 3.1algebra

Choose a finite $z_{0} \notin K^{'}$ and define $G (z) = (F_{0} (z) - F_{0} (z_{0})) / (z - z_{0})$ for $z \neq z_{0}$ , $G (z_{0}) = F_{0}^{'} (z_{0})$ , and $G (\infty) = a$ . Because $F_{0}$ is holomorphic near $z_{0}$ and has the expansion in step 3.1 near infinity, these values make $G$ continuous on the sphere and holomorphic near both $z_{0}$ and infinity. On $K^{'}$ , the denominator is nonzero and $F_{0}$ is finite because $F_{0}^{- 1} (\infty) = \infty$ ; hence $G$ is continuous there as well. Thus $G$ is holomorphic on $\hat{C} ∖ K^{'}$ .

5.1F3F4F5F9step 3.1step 4.1algebra

By [F9], the Countable Choice hypothesis of [F4] follows from AC. Apply [F4] to $G$ and $K^{'}$ ; then $G$ is constant, with value $a \neq 0$ . For every finite $z \neq z_{0}$ , including points of $K^{'}$ , the quotient identity gives $F_{0} (z) = F_{0} (z_{0}) + a (z - z_{0})$ ; continuity gives the same identity at $z_{0}$ . Therefore $F_{0}$ is an affine Möbius transformation. Since $F = T^{- 1} \circ S^{- 1} \circ F_{0} \circ T$ and Möbius maps form a group, $F$ is Möbius. As $F$ was arbitrary, [F5] proves that $K$ is globally conformally removable.

6.1F6F7F8given∎

Let $Γ$ be a quasicircle. By [F8], $Γ = h (S^{1})$ for a quasiconformal sphere homeomorphism $h$ . The round circle is globally conformally removable by [F6], so [F7] makes $Γ$ globally conformally removable. The exact supplier chains and their consuming uses in this step are recorded in [F6]–[F8].

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92 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Malik Younsi, On removable sets for holomorphic functions, EMS Surveys in Mathematical Sciences 2 (2015), 219–254 (standard reference, not scraped)
Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I (book draft, Stony Brook) (standard reference, not scraped)
Christopher J. Bishop, Quasiconformal Mappings (Stony Brook Math 627 course notes, 164 pp.) (standard reference, not scraped)