Non-Conformally Removable Sets

• Non-conformally removable sets are compact subsets of the complex plane where conformal or quasiconformal maps cannot extend globally as Möbius transformations.
• They are characterized through metric surface techniques, capacity theory, and Finsler metrics which highlight the failure of analytic continuation across the set.
• Examples such as Sierpiński carpets, Cantor set products, and flexible Jordan curves demonstrate their impact on rigidity, conformal welding, and the classification in geometric function theory.

A non-conformally removable set is a compact subset of the complex plane or the Riemann sphere for which there exists a homeomorphism, conformal (or quasiconformal) away from the set, that fails to extend as a global Möbius transformation. These sets precisely delimit the failure of analytic extension phenomena in geometric function theory, and their structure underpins rigidity, flexibility, and classification questions in conformal and quasiconformal geometry.

1. Definitions and Characterizations

A compact set $E \subset \widehat{\mathbb{C}}$ is called conformally removable if every homeomorphism $F: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ that is conformal off $E$ must globally be a Möbius transformation (i.e., extends conformally everywhere). This naturally generalizes to $S$-removability and $CH$-removability:

• S-removable: Every conformal embedding $f: \mathbb{C} \setminus E \to \widehat{\mathbb{C}}$ extends to a global conformal map.
• CH-removable: Every homeomorphism $f:\mathbb{C}\to\mathbb{C}$ conformal (or quasiconformal) off $E$ is globally conformal.

A set is non-conformally removable if it fails any of these conditions. Such non-removability is detected not just in the classical planar self-maps, but in the inability to collapse the set to zero measure via maps onto any metric surface (in the sense of locally finite Hausdorff 2-measure). This broadens the geometric perspective beyond Euclidean domains to include metric surface parametrizations (Ntalampekos, 2024, Ikonen et al., 2020).

2. Metric Surface Framework and Failure Conditions

The modern approach utilizes metric surfaces and reciprocity. The following central theorems encapsulate the metric-surface perspective (Ntalampekos, 2024):

• $E$ is S-removable if and only if there exists a quasiconformal homeomorphism onto some metric surface $X$, such that $\mathcal{H}^1(f(E))=0$.
• CH-removability is characterized similarly, with $\sigma$-finite linear measure and an additional reciprocity condition.

Non-removability requires that, for every metric surface parametrization $f$ of $\widehat{\mathbb{C}}$, $\mathcal{H}^1(f(E))>0$ — equivalently, no quasiconformal map to any metric surface can collapse $E$ to zero linear measure (Ntalampekos, 2024). This criterion precludes analytic extension across the set, regardless of the target geometry.

Further, in the Finsler-metric setting, non-removability is encoded by the non-reciprocity of the quotient space $(\Omega/\sim, d_N)$ where $N$ is a degenerate Finsler structure vanishing precisely on $E$ (Ikonen et al., 2020).

3. Explicit Examples and Constructions

Classical and contemporary literature provides a rich supply of non-removable sets.

• Sierpiński carpets and gaskets: Every planar Sierpiński carpet and the Sierpiński gasket are non-removable for (quasi)conformal maps. There exists a homeomorphism of the sphere which is conformal off the carpet (or gasket), mapping it onto another such set, but which is not globally conformal (Ntalampekos, 2018, Ntalampekos, 2018). In higher dimensions ($n \geq 2$), (n−1)-dimensional Sierpiński spaces within $S^n$ are non-removable: there exists a homeomorphism conformal outside the space mapping it to a set of positive measure, violating the necessary measure-preservation property of (quasi)conformal maps (Ntalampekos et al., 2018).
• Cantor set products: The product $E\times F$ of two Cantor sets can be non-removable if, for example, $F$ is sufficiently “thick” in logarithmic capacity so that $$\operatorname{cap}([a,b]\setminus F) < \operatorname{cap}([a,b])$$ (Rajala, 2024). This provides explicit classes of totally disconnected non-removable sets, including sets of the form $E\times [0,1]$.
• Zero-area Jordan curves and flexible curves: There exist flexible Jordan curves of zero area that are not conformally removable. Associated are explicit homeomorphisms of the sphere, conformal off the curve and mapping the curve onto itself, which are not Möbius (Younsi, 2017).
• Others: Giant families of non-removable sets also arise as images under conformal maps, e.g., complements of products of Cantor sets and those with particular tangency or unrectifiability properties (cf. Kaufman, Bishop).

4. Methodologies for Proving Non-removability

Proofs employ diverse tools, including:

• Topological subdivisions and gluing: For Sierpiński carpets, iterative $\varepsilon$-subdivisions and controlled conformal gluing (with prescribed behaviour on peripheral circles) yield the desired exceptional homeomorphisms (Ntalampekos, 2018).
• Collapse and folding maps into non-planar surfaces: Gasket cases use “collapse-and-fold” maps into metric surfaces (flap-planes), followed by quasisymmetric uniformization (e.g., Bonk-Kleiner’s theorem) to embed the result back into the plane. The resulting maps are quasiconformal off $E$ but cannot be quasiconformal globally due to the measure increase of $F(E)$ (Ntalampekos, 2018).
• Finsler metric and modulus arguments: For linear Cantor sets, considering degenerate weight functions vanishing on $E$ reveals the failure of reciprocity/extension manifested in the positivity of the modulus of certain families of curves post-collapse (Ikonen et al., 2020).
• Logarithmic capacity and analytic capacity tools: Non-removability of products is often established by capacity considerations, applying Ahlfors-Beurling results and analytic function-theoretic methods (Rajala, 2024).

5. Broader Implications and Open Problems

The existence of non-removable sets has significant implications:

• Rigidity and conformal welding: Non-removable sets obstruct conformal welding uniqueness; flexible (non-removable) curves exhibit non-uniqueness, undermining injectivity of the welding correspondence (Younsi, 2017). For circle domains, the existence of domains with non-removable boundary but rigidity (all conformal automorphisms are Möbius) illustrates subtleties in the rigidity versus removability dichotomy (Rajala, 2024).
• Topological versus geometric criteria: Topological structure — especially “infinite loop” systems or totally disconnectedness — can force non-removability, independent of geometric size or measure. There is no complete geometric characterization of removable sets; Hausdorff dimension and measure are not sufficient discriminants.
• Open directions: Central unresolved questions include the equivalence (or lack thereof) between quasiconformal and CH-removability in the plane, the explicit construction of concrete wild Cantor-type non-removable sets (with prescribed porosity or dimension), and a detailed understanding of the spectrum of removable versus non-removable sets with respect to geometric dimension (Ntalampekos, 2024, Ntalampekos, 2018).

6. Illustrative Summary Table

Set Class	Non-removability Mechanism	Reference
Sierpiński carpet/gasket	Topological subdivision + exceptional homeo	(Ntalampekos, 2018, Ntalampekos, 2018)
Sierpiński spaces in $S^n$	Shrinking-gluing limiting conformal homeo	(Ntalampekos et al., 2018)
Product Cantor sets $E\times F$	Capacity-theoretic failure	(Rajala, 2024)
Zero-area Jordan curve	Flexible curve + log-singular welding	(Younsi, 2017)

The significance of non-conformally removable sets is pervasive across planar and higher-dimensional analysis, impacting not only the structure and extension of conformal or quasiconformal maps, but also rigid structure questions, the geometry of fractal boundaries, and the analytic classification of compacta in complex dynamics and Teichmüller theory.