Proof. We give the contour construction from the component quotient, using Moore’s classical decomposition theorem. Collapse every component of \(F=\widehat{\mathbb C}\setminus V\) to a point, leaving domain points as singletons. This decomposition has closed equivalence relation. For convergent related pairs, a subsequence in domain singleton classes forces equal limits. Otherwise pass to a Hausdorff limit of complementary components; it is a continuum in \(F\) containing both limiting points, which therefore remain in the same component. Every component \(K\) is nonseparating. Indeed \(V\) lies in one component of \(\widehat{\mathbb C}\setminus K\); any other component \(A\) would miss \(V\), making \(K\cup\overline A\) a connected subset of \(F\) strictly larger than \(K\). Moore’s Decomposition Theorem (Moore 1925) therefore gives a closed monotone quotient \(\pi:\widehat{\mathbb C}\to X\) onto a topological sphere. Its restriction to \(V\) is a homeomorphism onto an open subset. The omitted set \(B_X\subset X\) is compact and totally disconnected: the inverse image of a continuum under a closed monotone map is connected, so a continuum in \(B_X\) lifts into a single component of \(F\).

For a physical Jordan curve \(C\subset V\) avoiding infinity, \(\pi|_C\) is a homeomorphism onto a Jordan curve. Every other decomposition element is connected and misses \(C\), so each physical Jordan side is saturated. Their images are therefore disjoint connected open sets partitioning \(X\setminus\pi(C)\), hence are its two Jordan sides; the bounded physical side maps to the side missing \(\pi(\infty)\). The quotient disks below are these images of bounded physical disks.

The compact metric space \(B_X\) has a clopen base. Partition it into finitely many clopen pieces in small quotient Jordan neighborhoods. Their physical inverse images have disjoint open neighborhoods avoiding the other pieces. Regular levels of smooth cutoffs around these compact inverse images give finitely many smooth curves in \(V\). Fill the outer curves, discard nested ones, and discard any resulting disk that misses the omitted set. Filling stays inside the chosen quotient Jordan neighborhood: the inverse image of its complement is connected, contains infinity, and misses the curve. It therefore lies on the unbounded side. This constructs finite families of physical Jordan disks with disjoint closures and smooth boundaries in \(V\), whose quotient images cover \(B_X\) by their interiors.

Refine these covers with new closures inside old disks and quotient mesh tending to zero. Their closed exteriors are connected smooth cores \(L_n\subset V\) with \[L_n\subset\operatorname{int}L_{n+1},\qquad \bigcup_nL_n=V.\] For \(b\in B_X\), let \(U_n(b)\subset X\) be its containing quotient disk and let \(C_n(b)=\pi^{-1}(\partial U_n(b))\subset V\) be the corresponding physical boundary. Then \[\overline{U_{n+1}(b)}\subset U_n(b),\qquad \bigcap_n\overline{U_n(b)}=\{b\}.\] The corresponding physical disks shrink to \(\pi^{-1}(b)\); their physical diameters need not tend to zero.

All these curves and cores lie in \(V\), where \(g\) is already defined. The compact embedded surface \(g(L_n)\) is the closed exterior of the disjoint bounded Jordan disks enclosed by its boundary curves. Orientation fixes their sides, since the interior of the core is connected and contains infinity. Let \(Q_n(b)\) be the closed bounded disk enclosed by \(g(C_n(b))\). Refinement gives \[Q_{n+1}(b)\subset\operatorname{int}Q_n(b),\qquad K'_b=\bigcap_nQ_n(b).\] This intersection is a nonempty continuum missing \(V'\), because the image cores exhaust \(V'\). Conversely every point outside \(V'\) lies in one image disk at each level. The corresponding nested source disks determine a unique \(b\) by their quotient mesh. A connected subset of the target complement cannot cross an image contour, so \(K'_b\) is exactly a complementary component. This proves the bijection without any countability assumption.

Given an open \(W\supset K'_b\), nested compactness gives \(Q_n(b)\subset W\) for sufficiently large \(n\). The physical inverse image of \(U_n(b)\) is an open neighborhood of \(\pi^{-1}(b)\), and every fiber over this neighborhood lies in \(Q_n(b)\). This proves upper semicontinuity. Applying the same argument to \(g^{-1}\) proves the inverse assertion. Singleton source fibers then give the stated continuous collapse map. When both sets of fibers are singletons, the extension is a continuous bijection of compact spheres and hence a homeomorphism. Its orientation is fixed by its restriction to \(V\). ◻

Proof. Apply the Exterior Area Theorem (Ahlfors 1973) to \(h(R_0\zeta)/R_0\). For \(h(z)=z+\sum_{n\ge1}\alpha_nz^{-n}\) it gives \[\sum_{n\ge1}n|\alpha_n|^2R_0^{-2n}\le R_0^2.\] Thus \(|b(h)|\le R_0^2\), and Cauchy–Schwarz gives the asserted uniform exterior estimate. In particular \(h(\partial D_{2R_0})\) lies in a fixed disk. Its bounded side contains the whole omitted set and \(h(G\cap D_{2R_0})\): injectivity prevents an interior image point from crossing this image circle, and points just inside the circle lie on its bounded side. Thus the maps are uniformly bounded on \(G\cap D_{2R_0}\) as well as on exterior compact sets, proving local normality. The uniform exterior estimate prevents a constant limit and preserves the normalization. Hurwitz’s Theorem then makes every limit univalent. ◻

Proof. If a removable compact set had positive area, choose a bounded measurable subset \(A\) of positive planar area in a coordinate chart and a coefficient \(\mu=k\mathbf1_A\) with \(0<k<1\). The Measurable Riemann Mapping Theorem (Ahlfors 2006) gives an orientation-preserving quasiconformal sphere homeomorphism with this Beltrami coefficient. It is conformal off the compact set and is not Möbius, since its coefficient is nonzero on a set of positive area. This is a contradiction. For the second assertion, a homeomorphism conformal off a smaller compact set is also conformal off the larger one, so the defining removability implication applies. ◻

Proof. For an integer \(n\ge2\), set \(k=n\) on \(F\) and \(k=1\) elsewhere. We first construct the principal quasiconformal coordinate \(\phi_n=a_n+iv_n\) satisfying \[ (a_n)_X=k(v_n)_Y,\qquad (a_n)_Y=-k(v_n)_X, \qquad \phi_n(w)=w+\frac{c_n}{w}+O(w^{-2}). \tag{10}\] Indeed this system is equivalent to the real-linear Beltrami equation \[(\phi_n)_{\bar w}=q\,\overline{(\phi_n)_w}, \qquad q=\frac{k-1}{k+1}.\] We use the classical Cauchy–Beurling construction of the Beltrami solution (Ahlfors and Bers 1960, secs. 1–3). Let \(\mathcal C\) be the plane Cauchy transform and \(\mathcal S=\partial\mathcal C\) its \(L^2\)-isometric Beurling transform, with \(\bar\partial\mathcal C g=g\). The equation \[g=q\,\overline{1+\mathcal Sg}\] is an affine contraction on \(L^2(\mathbb C)\) considered as a real Banach space. Its solution is supported on \(F\). Thus \(\phi_n=w+\mathcal Cg\) is locally Sobolev and has the expansion in (10); compact support also makes \(g\) integrable. Set \(\mu=(\phi_n)_{\bar w}/(\phi_n)_w\) where the denominator is nonzero, and \(\mu=0\) otherwise. The distortion inequality \(|(\phi_n)_{\bar w}|\le\|q\|_\infty|(\phi_n)_w|\) shows that \((\phi_n)_{\bar w}=\mu(\phi_n)_w\) almost everywhere, \(\|\mu\|_\infty<1\), and \(\mu=0\) off \(F\). Standard planar weak-solution regularity, the Measurable Riemann Mapping Theorem, and holomorphic factorization (Ahlfors and Bers 1960; Ahlfors 2006; Astala et al. 2009) give a continuous representative \(\phi_n=H\circ\psi\), where \(\psi\) is the principal quasiconformal plane homeomorphism solving \(\psi_{\bar w}=\mu\psi_w\) and \(H\) is entire. Since \(\psi(w)=w+O(1/w)\) and \(\phi_n\) has the displayed principal expansion, \(H(w)=w+O(1/w)\) near infinity. Liouville’s Theorem makes \(H\) the identity. Consequently \(\phi_n\) itself is a homeomorphism. Direct differentiation gives the signs and the Jacobian: \[ (\phi_n)_w=\frac{k+1}{2}\bigl((v_n)_Y+i(v_n)_X\bigr),\quad (\phi_n)_{\bar w}=\frac{k-1}{2}\bigl((v_n)_Y-i(v_n)_X\bigr),\quad J_{\phi_n}=k|\nabla v_n|^2. \tag{11}\]

The map \(\phi_n\circ h\) is again a model. It is conformal on \(G\), and its new coefficient is \(b(h)+c_n\). A homeomorphism carries complementary components to complementary components, preserving points and Jordan disks; uniform continuity on a compact set containing \(F\) preserves the null-family property. For an old continued test \(U\), its new continuation is exactly \(U\circ\phi_n^{-1}\), with the same prescribed constant on the corresponding component. Quasiconformal composition preserves its global \(W^{1,2}(\widehat{\mathbb C})\) membership and its zero gradient on the transformed complement. To see also the null-set issue in this statement, approximate the continuous \(U\) uniformly and strongly in \(W^{1,2}\) by smooth sphere functions. The quasiconformal Dirichlet inequality makes their compositions converge in gradient norm, and uniform convergence identifies the limit. The almost-everywhere chain rule then applies. The forward Lusin N property of \(\phi_n\) maps the source-null exceptions to target-null sets, so the resulting gradient vanishes almost everywhere on \(\phi_n(F)\). Approximate maximality therefore gives \[ \Re c_n\le d. \tag{12}\]

For \(D\) larger than a disk containing \(F\), the area formula and the boundary Laurent expansion give \[\int_{D_D}J_{\phi_n}\,\mathrm dA =\mathop{\mathrm{area}}\bigl(\phi_n(D_D)\bigr)=\pi D^2+o(1) \quad(D\to\infty).\] Here \(D_D=\{w:|w|<D\}\). The Divergence Theorem, or the weak boundary formula on this smooth exterior circle, gives the exact identity \[\int_{D_D}(v_n)_Y\,\mathrm dA =\int_{\partial D_D}v_n\,\nu_Y\,\mathrm ds =\pi D^2-\pi\Re c_n;\] all other Laurent terms have zero contribution. Expanding the square, using (11), and then letting \(D\to\infty\) yields \[ \int_{\mathbb C}|\nabla(v_n-Y)|^2\,\mathrm dA +(n-1)\int_F|\nabla v_n|^2\,\mathrm dA =2\pi\Re c_n\le2\pi d. \tag{13}\] In particular both terms on the left are controlled, with the stated normalization constant.

We next obtain compactness without a uniform quasiconformal distortion bound. A coordinate of a plane homeomorphism attains its maximum and minimum on the boundary of every relatively compact open set: an interior coordinate extremum contradicts openness. For almost every circle centered at \(x\), the Sobolev circle restriction and Cauchy–Schwarz therefore give, for \(0<r<R\), \[ \bigl(\mathop{\mathrm{osc}}_{\overline{D(x,r)}}v_n\bigr)^2\log(R/r) \le 2\pi\int_{D(x,R)}|\nabla v_n|^2\,\mathrm dA. \tag{14}\] One obtains this inequality by bounding the oscillation on the smaller disk by that on a circle of radius \(t\in(r,R)\), bounding the latter by the integral of \(|\nabla v_n|\) around that circle, and integrating with respect to \(\mathrm dt/t\). Equation (13) bounds the right-hand side locally, uniformly in \(n\). Exterior univalence and principal normalization, on a common exterior disk, give \[ |v_n(w)-Y|\le C/|w|\qquad(|w|\ge R_0), \tag{15}\] where \(C,R_0\) depend only on the common bound for \(F\), hence only on \(B\). The maximum and minimum principles give uniform bounds on compact sets as well. Thus (14) gives local equicontinuity.

Pass to a locally uniform and locally Sobolev-weak subsequential limit \(v\). Lower semicontinuity proves (9), and the second term of (13) forces \(\nabla v=0\) almost everywhere on \(F\). The function is therefore constant on the connected interior of each Jordan disk component, and continuity extends the same value to its boundary. Singleton components need no argument. The coordinate maximum and minimum principles and the exterior estimate pass to the limit.

Finally, allow the nearly maximizing models to vary and let \(d\to0\). The same modulus and exterior bounds give compactness on every compact set. Any subsequential limit of \(v-Y\) has zero distributional gradient by (9), hence is constant. Estimate (15), with arbitrarily large \(|w|\), makes the constant zero. This proves the uniform compact-set closeness asserted in the proposition. ◻

Proof. For the approximating homeomorphism \(\phi_n\), the inverse image of the horizontal line through \(\phi_n(x)\) is a proper simple curve through \(x\). Stop one branch at its first exit from the prescribed ball. The resulting compact arcs have a Hausdorff convergent subsequence. Their limit is connected, contains \(x\), and meets the ball boundary. Local uniform convergence \(v_n\to v\) places it in the actual level \(v^{-1}(v(x))\). This proves the first assertion pointwise.

We use the scalar Sobolev Coarea Theorem, due to Federer (Federer 1969), in the precise-representative form of Malý, Swanson, and Ziemer (Malý et al. 2003, Theorem 1.1). With domain dimension \(2\), target dimension \(1\), and exponent \(2\), it gives, for every nonnegative Borel weight \(a\) supported in a bounded region, \[ \int_{\mathbb R}\int_{v^{-1}(t)}a\,\mathrm d\mathcal H^1\,\mathrm dt =\int_{\mathbb C}a|\nabla v|\,\mathrm dA. \tag{17}\] Continuity makes \(v\) equal everywhere to its precise representative, so these are the full pointwise level sets. In particular, almost every level has finite length in any fixed compact box. Formula (17) also implies that planar null sets have zero length on almost every level.

Choose a closed box \(Q\) with \(F\) in its interior. On a neighborhood of \(Q\), smooth approximation of the continuous Sobolev function \(U\) provides \(U_j\to U\) uniformly on \(Q\) with \[\sum_j\|\nabla U_j-\nabla U\|_{L^2(Q)}<\infty.\] Let \(g\) be a nonnegative Borel representative of \(|\nabla U|\). By (17) and Cauchy–Schwarz, outside one null set of values \(t\) we have simultaneously \[\begin{gather*} \mathcal H^1(Q\cap v^{-1}(t))<\infty,\qquad \int_{Q\cap v^{-1}(t)}g\,\mathrm d\mathcal H^1<\infty,\\ \sum_j\int_{Q\cap v^{-1}(t)} |\nabla U_j-\nabla U|\,\mathrm d\mathcal H^1<\infty, \qquad \int_{F\cap v^{-1}(t)}g\,\mathrm d\mathcal H^1=0. \end{gather*}\] The last assertion uses \(g=0\) almost everywhere on \(F\); the coarea identity makes the choice on the planar-null exceptional set irrelevant. For every such good level and every rectifiable simple arc \(\gamma\subset Q\cap v^{-1}(t)\), the smooth arc inequality passes to the limit to give \[ |U(x)-U(y)|\le\int_\gamma g\,\mathrm d\mathcal H^1 \qquad(x,y\text{ the endpoints of }\gamma). \tag{18}\] Indeed the gradient error on any arc is bounded by the error on the entire level section, whereas uniform convergence controls both endpoints. Thus the exceptional levels do not depend on the choice of arc or its endpoints.

Fix a good level. Its nondegenerate components in \(Q\) form an at most countable family of finite-length continua. Each such continuum is locally connected: failure of local connectedness, by boundary bumping, would give infinitely many disjoint subcontinua of diameter bounded below, contrary to finite length. A locally connected metric continuum is locally arcwise connected, so its connected relatively open subsets are arcwise connected. All their simple arcs are rectifiable, since their lengths are bounded by the length of the whole continuum.

Let \(K\) be one of these components and \(A=F\cap K\). The finite measure \(g\mathcal H^1|_K\) vanishes on \(A\). Given \(\epsilon>0\), choose a relatively open neighborhood \(O\) of \(A\) in \(K\) with \(\int_Og\,\mathrm d\mathcal H^1<\epsilon\). Local connectedness and second countability make its connected components \(O_i\) a countable family of relatively open sets. Joining any two points of \(O_i\) by a simple arc there and using (18) shows \[\mathop{\mathrm{diam}}U(O_i)\le\int_{O_i}g\,\mathrm d\mathcal H^1.\] Since each \(U(O_i)\) is an interval, the outer length of \(U(A)\) is at most \(\epsilon\), and hence is zero. Every point of \(F\cap v^{-1}(t)\) belongs to a nondegenerate component of \(Q\cap v^{-1}(t)\) by the first assertion of the lemma, using a small ball inside \(Q\). Thus the countable family of components covers every such point, and \(U(F\cap v^{-1}(t))\) has length zero. The compact set \((U,v)(F)\) now has almost every horizontal section of length zero. Fubini proves its area is zero.

For the companion pair, the continuous Sobolev representative has the absolute continuity on lines (ACL) property. Thus \(U\) is absolutely continuous on almost every vertical line. Fubini gives \(\partial_YU=0\) almost everywhere on that line’s intersection with \(F\). The one-dimensional area inequality for an absolutely continuous function therefore makes the image of this intersection length-null. Applying Fubini to the compact set \((U,X)(F)\) proves the second equality in (16). ◻

Proof. We construct a continuous pair \(P=(p,q)\) and obtain the barriers by composing it with translates of a radial cutoff of small Dirichlet energy. By conformal invariance, composing the cutoff with a model map \(h\) does not increase its Dirichlet energy, so we will compare \(P\) with \((\Re h,\Im h)\) on a fixed source disk. Proposition 8 supplies a height \(v\) whose pullback gives the vertical coordinate. We make small range compressions of this pullback and the old tests so that they become constant near every source end corresponding to a nonpoint model hole, then modify the horizontal coordinate there. These modifications preserve pair-nullity and make the gradient error small on the source disk. The vertical error is uniformly small; the horizontal changes are confined to neighborhoods where \(h\) has small energy.

Choose \(a\ge1\) with \(B\subset D_a\). For every normalized univalent map \(h\) on \(G\), the Exterior Area Theorem gives \[ |h(z)-z|\le a\sqrt{-\log(1-a^2/|z|^2)}\qquad(|z|>a). \tag{20}\] Indeed, apply Cauchy–Schwarz to the Laurent series and the area-theorem coefficient bound. Hence \(h(\partial D_{2a})\subset D_{3a}\), and the omitted set \(F=\mathbb C\setminus h(G)\) lies on its bounded side. In particular, \(F\subset D_{3a}\) for every model of every list. Fix \(D_0=64a\). The image of \(G\cap D_{65a}\) lies in \(D_{67a}\); these uniform bounds also bound the energy of \((\Re h,\Im h)\) on that source disk.

Choose a smooth radial cutoff \(\Phi\), equal to one on a disk of radius \(r>0\) and zero outside \(D_{4a}\), with \(\|\nabla\Phi\|_{2,\mathbb C}<\eta/3\). Such a cutoff is obtained by smoothing a logarithmic profile and taking \(r\) sufficiently small. All its translates have the same derivative bounds. We shall use translates centered at the eventual traces of \((p,q)\) on \(B\).

By Proposition 8, take a model \(h\) sufficiently close to the supremum of its real Laurent coefficient that the associated height \(v\) satisfies \[ \|v-Y\|_{\infty,D_{67a}}\ \hbox{and}\ \|\nabla(v-Y)\|_{2,\mathbb C} \quad\hbox{as small as required below}, \qquad Y(w)=\Im w. \tag{21}\] Write \(U_i\) for the global model continuation of \(u_i\). The function \(v\) is constant on every component of \(F\), so \(v\circ h\) extends continuously to \(B\) by Lemma 2.

Flattening the hole values. For a continuous, locally Sobolev real function \(s\) of finite Dirichlet energy on a region, the measure \[\mu_s(I)=\int_{\{s\in I\}}|\nabla s|^2\] has no atom at any real number: the Sobolev gradient is zero almost everywhere on each fixed level set. Thus, given countably many values, we can surround them by open intervals whose total length and total \(\mu_s\)-measure are arbitrarily small. If \(O\) is their union, then \[T(t)=\int_0^t\mathbf1_{\mathbb R\setminus O}(\tau)\,\mathrm d\tau\] is \(1\)-Lipschitz, satisfies \(\|T-\mathrm{id}\|_\infty\le |O|\), and is constant near each specified value. The chain rule bounds the squared gradient error by \(\mu_s(O)\). This remains valid when the specified values are dense.

Apply this construction to each \(u_i\), at its values at the source ends corresponding to nonpoint components of \(F\), and to \(v\) at its values on those components. For \(v\) use only the energy on \(D_{67a}\). Denote the flattened old tests by \(\widetilde u_i=T_i\circ u_i\), and put \(q_0=T\circ v\circ h\). Their errors can be arbitrarily small, and the required bounds by \(\delta\) hold for the old tests. In particular, \(q_0\) is uniformly and energetically close to \(\Im h\) on \(G\cap D_{65a}\). By Lemma 9 and coordinatewise Lipschitz composition, the traces of \(q_0\) form null-area pairs with the flattened old tests. The old pairs also remain null. Each nonpoint-hole end \(b_j\) now has a source neighborhood on which all \(\widetilde u_i\) and \(q_0\) are constant.

The horizontal coordinate. Let \(F_j\) be the nonpoint component corresponding to \(b_j\), and choose \(c_j\in\Re F_j\). Around each \(b_j\) choose a ball \(B_j\subset D_{2a}\) in the preceding common constancy neighborhood. Its radius may be required to tend to zero and its image in \(G\) to lie within \(2^{-j}a\) of \(F_j\). The balls can also satisfy \[ \|\mathbf1_S\nabla H\|_2<\tau, \qquad S=\bigcup_jB_j,\qquad H=(\Re h,\Im h), \tag{22}\] for any prescribed \(\tau>0\): the energy is integrable, and each center is a single source point. Choose logarithmic cutoffs \(\chi_j\) supported in \(B_j\), equal to one near \(b_j\), with \(\sum_j\|\nabla\chi_j\|_2<\tau\). Put \[ w_j=\chi_j\prod_{i<j}(1-\chi_i),\qquad \rho=\prod_j(1-\chi_j),\qquad p_0=\rho\Re h+\sum_jw_jc_j\quad\hbox{on }G. \tag{23}\] These expressions are locally finite in \(G\), since the radii tend to zero and all centers belong to \(B\). They are convex interpolations. On \(S\), all horizontal values used have absolute value at most \(4a\). The product rule, applied to the finite formula locally, gives \[ |\nabla p_0-\rho\nabla\Re h| \le 8a\sum_j|\nabla\chi_j|,\qquad \|\nabla(p_0-\Re h)\|_{2,G\cap D_{65a}} \le(1+8a)\tau. \tag{24}\] No disjointness of the balls is required.

At \(b_j\), the factor \(\chi_j=1\) makes (23) terminate after index \(j\) on a neighborhood. It is therefore a finite continuous expression there. At a point end \(b\), write \(F_b=\{\zeta_b\}\) and \(x_b=\Re\zeta_b\). Early nonzero weights may change the trace from \(x_b\). To prove continuity, compare \(p_0\) with the finite continuous function \[x_b+\sum_{j\le N}w_j(z)(c_j-x_b).\] For sufficiently large \(N\) and sufficiently small neighborhoods of \(b\), every later ball meeting that neighborhood has its center near \(b\). Upper semicontinuity of the end correspondence then puts its entire component \(F_j\), and hence \(c_j\), arbitrarily close to \(\zeta_b\) and \(x_b\). Also \(\Re h(z)\to x_b\) from \(G\). The remaining weight is a convex combination of these close values, so the difference from the displayed finite expression is uniformly arbitrarily small. This proves continuous extension at \(b\), including when early supports overlap. The same estimate controls the prescribed values at nearby ends.

The new pairs are now checked on all of \(B\). On each ball \(B_j\), every coordinate other than \(p_0\) is constant, so a pair involving \(p_0\) has image in a line. Countably many balls contribute zero planar area. Outside \(S\), all ends are point ends and \(p_0\) is their horizontal trace. Apply the horizontal assertion of Lemma 9 to \(T_i\circ U_i\) and to \(T\circ v\), respectively. Their gradients vanish on \(F\), so this proves nullity of the pairs with every old test and with \(q_0\). Together with the earlier pair checks, this proves the full pair-null condition. Multiply \(p_0,q_0\) by a smooth source cutoff equal to one on \(D_{65a}\) and zero outside a larger disk. The resulting \(p,q\) are bounded continuous tests of finite energy on \(G\); no trace or working-disk calculation changes.

Small energy and strict margins. Let \(P=(p,q)\) on the working disk. Outside \(S\) its first coordinate equals that of \(H\), and write \(e=\|q-\Im h\|_\infty\) there. For any translate \(\Phi_l\) of \(\Phi\), with derivative bounds \(M_1=\|D\Phi\|_\infty\) and \(M_2=\|D^2\Phi\|_\infty\), the chain rule gives \[\begin{align*} \|\nabla\Phi_l(P)-\nabla\Phi_l(H)\|_2 &\le M_1\|\nabla P-\nabla H\|_2 +M_2 e\|\nabla H\|_2 +2M_1\|\mathbf1_S\nabla H\|_2 . \tag{25}\end{align*}\] All norms here are on \(G\cap D_{D_0}\). The first and last terms tend to zero by the choices above; the middle term does so by (21) and flattening. This estimate does not require a small uniform horizontal error on \(S\). Conformal invariance gives \(\|\nabla\Phi_l(H)\|_2<\eta/3\), so the choices can ensure the strict bound in (19), uniformly over all translates.

Take the total vertical uniform error less than \(a\). On \(B\), \(|p|\le3a\) by convexity and \(|q|\le4a\), so every trace center has norm at most \(5a\). On the collar \(63a\le |z|\le65a\), the first coordinate is \(\Re h\) and the second differs from \(\Im h\) by less than \(a\). Equation (20) puts its distance from every trace center above \(50a\), well beyond the support radius \(4a\). At each \(b\in B\), the translate centered at \(P(b)\) is one wherever \(|P-P(b)|<r\). Choose a smaller source neighborhood whose closure has \(|P-P(b)|<r/2\). Compactness and the contour construction in the proof of Lemma 2 give finitely many inner contours in such neighborhoods with bounded sides covering \(B\).

Finally, for a fixed saved pair \(A=(p,q)\) and any perturbed pair \(A'\), \[ \|\nabla\Phi_l(A')-\nabla\Phi_l(A)\|_2 \le M_1\|\nabla A'-\nabla A\|_2 +M_2\|A'-A\|_\infty\|\nabla A\|_2. \tag{26}\] The positive plateau and zero-region margins preserve the exact values one and zero under small uniform changes. The positive energy slack and (26) preserve the strict norm bound. This proves the stability assertion and the lemma. ◻

Proof of Theorem 10. The empty case is immediate. Fix \(0<\epsilon<1\) and start with the one-element list \(u\), for which pair-nullity is vacuous. Apply Lemma 11 successively with \(\eta=1/k\), saving at stage \(k\) its pair, curves, and functions \(\Phi_{k,l}\). By the strict margins and (26), there is \(t_k>0\) such that total uniform and gradient changes of that pair below \(t_k\) preserve all its saved conclusions with norm less than \(1/k\).

At stage \(n\), require each earlier scalar coordinate to change by less than \[d_n=\frac{2^{-n-3}}{\sqrt2} \min(1,\epsilon,t_1,\ldots,t_{n-1})\] in both norms (omit the \(t\)’s when \(n=1\)). Each stage has only finitely many requirements, and the lemma permits arbitrarily small changes. For every fixed \(k\), the sum of subsequent pair errors is less than \(t_k/4\). Thus all saved barriers work at every later finite stage. If \(a_n\) is the first coordinate at stage \(n\), then \[ a_n\longrightarrow u_\epsilon\text{ uniformly},\qquad \|u_\epsilon-u\|_\infty<\epsilon,\qquad \|\nabla a_n\|_{2,G}\le\|\nabla u\|_{2,G}+1. \tag{27}\] Only finite lists are required to be pair-null. In particular, no infinite-list model assertion or limiting pair-null assertion is being used.

Choose a model \(h_n\) of each resulting finite list by Proposition 6. Lemma 3 gives a subsequence converging locally uniformly on \(G\) to a normalized univalent map \(h\). We claim that every component of its omitted set is a point. Otherwise, by Lemma 2, some source end \(b\) has a nondegenerate target component \(K\). One axial projection of \(K\) contains an interval \(J\) of positive length.

For each \(k\), choose a saved inner contour \(\gamma_{k,l}\) enclosing \(b\). Its limiting image strictly encloses \(K\). Uniform convergence on the contour and winding-number stability imply that \(h_n(\gamma_{k,l})\) also encloses \(K\) for all sufficiently large \(n\) in the subsequence. Use the two continued model coordinates at stage \(n\) in \(\Phi_{k,l}\), on the bounded side of \(h_n(\partial D_{D_0})\), and extend by zero outside. Denote this function by \(V_{n,k}\). It is continuous and globally Sobolev: the saved zero collar removes any interface jump, and the model coordinates are globally Sobolev. Their gradients vanish on the model holes. The chain rule and conformal invariance therefore give \[ \|\nabla V_{n,k}\|_{2,\mathbb C}<1/k. \tag{28}\] It equals one on \(h_n(\gamma_{k,l})\). Moreover all such functions are zero outside one fixed disk \(D_L\), for example with \(L=67D_0/64\) by the uniform bounds in the preceding proof. The radius does not depend on \(k\), \(n\), or the coordinate list.

For definiteness suppose \(J\) is the vertical projection interval. Every horizontal line at a height in \(J\) meets the inner image contour, where \(V_{n,k}=1\). On almost every such line, the Sobolev restriction is absolutely continuous and agrees with the continuous restriction. It takes value zero at the left and right ends of the segment with horizontal coordinates \(-L\) and \(L\). Cauchy–Schwarz, followed by integration over \(J\), gives \[ \int_\mathbb C|\nabla V_{n,k}|^2 \ge\frac{|J|}{2L}>0. \tag{29}\] The argument only uses the value one on the contour, as illustrated in Figure 1. Choose \(k\) so large that \(k^{-2}<|J|/(2L)\), and then choose \(n\) sufficiently late for that saved contour. Equations (28) and (29) contradict each other.

All limit components are therefore singletons. Lemma 2 now extends \(h\) to an orientation-preserving homeomorphism of the sphere. Removability of \(B\) makes it a Möbius map; the normalization \(h(z)=z+O(1/z)\) makes it the identity.

Let \(A_n\) denote the globally continued first test of the model \(h_n\). Its values on holes are the source endpoint values, and its gradient vanishes there. Consequently \[\|A_n\|_\infty\le\|u\|_\infty+1,\qquad \|\nabla A_n\|_{2,\widehat{\mathbb C}} =\|\nabla a_n\|_{2,G}\le\|\nabla u\|_{2,G}+1.\] Take a weakly convergent subsequence in \(W^{1,2}(\widehat{\mathbb C})\). Since \(h_n\to\mathrm{id}\) locally uniformly on \(G\), inverse convergence and (27) give \(A_n\to u_\epsilon\) locally uniformly on \(G\). Thus the weak limit equals \(u_\epsilon\) there. Lemma 4 gives \(|B|=0\), so the equality holds almost everywhere on the sphere. This proves global Sobolev membership of \(u_\epsilon\), with a bound independent of \(\epsilon\). Finally let \(\epsilon\downarrow0\) and use uniform convergence to \(u\) and weak Sobolev compactness once more. This proves the theorem. ◻

Proof. Let \(g\geq0\) belong to \(L^2\), and let \(Mg\) be its centered maximal function. For every \(x\in B_j\), the average of \(g\) on \(B_j\) is at most \(4Mg(x)\), since \(B_j\subset B(x,2d_j)\). Consequently \[\begin{align*} \int g\sum_j\frac{a_j}{d_j}{\bf1}_{B_j} &\leq 4\pi\sum_j a_jd_j\inf_{E_j}Mg\\ &\leq C_c\left(\sum_j a_j^2\right)^{1/2} \left(\sum_j\int_{E_j}(Mg)^2\right)^{1/2} \leq C_c\|(a_j)\|_{\ell^2}\|g\|_2. \end{align*}\] Here the last inequality uses disjointness and the \(L^2\) Maximal Theorem. Duality proves the assertion for finite sums, and monotone convergence proves it for countable sums. ◻

Proof. We fix \(e\) throughout. All complementary components on either side lie in bounded planar regions. Index those pairs \((K_j,K'_j)\) for which at least one component is a disk of positive radius. The index set is finite or countable, since disjoint positive-radius disks have disjoint nonempty interiors. Write \[K_j=\overline B(a_j,r_j),\qquad K'_j=\overline B(a'_j,s_j),\] allowing radius zero, and set \(\beta_j=\pi_e(a'_j)\). Disjointness and boundedness give \[ \sum_jr_j^2<\infty,\qquad \sum_js_j^2<\infty. \tag{30}\] The other components are precisely the common points in \(C\).

Scalar data and shrinking neighborhoods. Choose a smooth planar cutoff \(\chi\), equal to one on a sufficiently large disk containing the source complement and all the neighborhoods introduced below, with its transition region compactly contained in \(\Omega\). On \(\Omega\) put \[b=\chi\,\pi_e(f),\] and extend \(b\) by \(\beta_j\) on \(K_j\), by \(\pi_e(f_*(x))\) on \(C\), and by zero at infinity. This is bounded. Indeed, \(f(\infty)=\infty\) implies that the image of a bounded source region is bounded. The area formula for the univalent map \(f\) then shows that \(b\) has finite Dirichlet energy on \(\Omega\); the cutoff-derivative term is supported in a compact subset of \(\Omega\).

Although \(b\) need not be continuous at \(K_j\), it is continuous at every common point. Moreover, the cluster values near \(K_j\) lie in \([\beta_j-s_j,\beta_j+s_j]\). These statements include approaches through other complementary components and follow from the upper-semicontinuous correspondence in Lemma 2.

Choose \(\varepsilon_j>0\) tending to zero with \(\sum_j\varepsilon_j^2<\infty\), and put \[I_j=(\beta_j-s_j-\varepsilon_j,\, \beta_j+s_j+\varepsilon_j),\qquad l_j=|I_j|.\] Thus \((l_j)\in\ell^2\). Fix \(A\) so large that all these intervals and all values of \(b\) lie in \((-A,A)\). Choose positive \(\Delta_j\) with \[ \sum_{i>j}\Delta_i^2<\frac12\Delta_j^2; \tag{31}\] for example, a sufficiently small multiple of \(2^{-j}\) suffices. For each integer \(m\geq1\), define the balloons \[B_{j,m}=B(a_j,d_{j,m}),\qquad d_{j,m}= \begin{cases} (1+1/m)r_j,&r_j>0,\\ \Delta_j/m,&r_j=0. \end{cases}\] These balls are all in one bounded region, and decrease to \(K_j\) as \(m\to\infty\). Choose radial Lipschitz functions \(0\leq\lambda_{j,m}\leq1\) that vanish on an open neighborhood of \(K_j\), equal one outside \(B_{j,m}\), and satisfy \[ |\nabla\lambda_{j,m}| \leq\frac{C_m}{d_{j,m}}{\bf1}_{B_{j,m}} \quad\hbox{almost everywhere}. \tag{32}\] One may take \(C_m=2(m+1)\), interpolating radially between \((1+1/(2m))r_j\) and \((1+1/m)r_j\) when \(r_j>0\), and between \(d_{j,m}/2\) and \(d_{j,m}\) otherwise.

For the moment fix \(m\) and suppress it from the notation. We will transform the scalar value \(b(z)\) before applying Sobolev extension. Near \(K_j\) the transformation should be constant on \(I_j\), so it does not distinguish the possible cluster values of \(b\). The cutoff \(\lambda_j\) imposes no restriction outside \(B_j\). Taking the infimum of the cutoffs whose intervals contain a given scalar value enforces these requirements simultaneously, including where the intervals and neighborhoods overlap. Integrating in the scalar variable makes the spatial cost proportional to \(l_j\), as the estimate below will show. Define \[ L(z,t)=\inf_{j:\,t\in I_j}\lambda_j(z),\qquad F(z,t)=\int_{-A}^{t}L(z,s)\,\mathrm ds,\qquad U(z)=F(z,b(z)), \tag{33}\] with the empty infimum equal to one. We shall prove that \(U\) is continuous and belongs to \(W^{1,2}(\widehat{\mathbb C})\).

An energy density. Set \[ P=C_m\sum_j\frac{l_j}{d_j}{\bf1}_{B_j}. \tag{34}\] This belongs to \(L^2(\mathbb C)\). For the positive-radius sublist, the interiors of the original disks \(K_j\) are disjoint subsets of \(B_j\) of area at least \(\pi d_j^2/4\), so Lemma 12 applies. For the zero-radius sublist use instead \[E_j=B_j\setminus\bigcup_{i>j,\,r_i=0}B_i.\] These sets are pairwise disjoint: if \(i<j\), then \(E_i\) excludes \(B_j\supset E_j\). Since all zero-radius balloons in this system use the same factor \(1/m\), (31) gives \[|E_j|\geq\pi d_j^2-\pi\sum_{i>j,\,r_i=0}d_i^2 \geq\tfrac12\pi d_j^2.\] Apply Lemma 12 separately to this sublist and then add the two \(L^2\) bounds. Overlap between the two sublists is irrelevant.

Continuity of the masks. We spell out the accumulation issue. For fixed \(m\), \(d_j\to0\). If distinct balloon supports accumulate at a common point \(x\), their centers and source components approach \(x\). By Lemma 2 their target components approach \(\{f_*(x)\}\); as \(\varepsilon_j\to0\), the corresponding intervals \(I_j\) approach the scalar \(b(x)\). Consequently, given \(\delta>0\), there are a neighborhood \(V\) of \(x\) and a finite initial list such that every remaining balloon meeting \(V\) has its interval in \((b(x)-\delta,b(x)+\delta)\). The finite-mask version of \(F\) differs from \(F\) by at most \(2\delta\) on \(V\), uniformly in the upper endpoint. Since the finite-mask version is continuous in its two variables and \(b\) is continuous at \(x\), \(U\) is continuous there.

For an indexed component \(K_j\), choose a neighborhood \(V_j\) of the entire closed component on which \(\lambda_j=0\) and \(b\in I_j\). This is possible because \(I_j\) has a positive margin around all its cluster values. On \(V_j\) the upper endpoint in (33) may be replaced by the fixed value \(\beta_j\). All sufficiently late intervals of balloons meeting a sufficiently small neighborhood of \(K_j\) lie in \(I_j\). To verify the uniformity of this assertion, a failure would give \(i_\nu\to\infty\) and \(z_\nu\in B_{i_\nu}\) with \(\mathop{\mathrm{dist}}(z_\nu,K_j)\to0\), but \(I_{i_\nu}\not\subset I_j\). Passing to a subsequence gives \(z_\nu\to x\in K_j\); the shrinking radii imply \(K_{i_\nu}\to\{x\}\). Component correspondence puts their target components arbitrarily close to \(K'_j\), and the vanishing extra margins then force \(I_{i_\nu}\subset I_j\), a contradiction. These late intervals have no effect on \(L\) near \(K_j\): there \(\lambda_j=0\), so \(L(z,t)=0\) for \(t\in I_j\). Thus near \(K_j\) the fixed-endpoint expression is a finite continuous mask. This proves continuity there, including when \(r_j=0<s_j\).

In \(\Omega\) the cutoffs are locally finite, and near infinity they are all one. Hence the remaining continuity assertions are immediate, with \(U=A\) near infinity.

Sobolev regularity and the residual dust. Let \(L_N,F_N,U_N\) denote the masks using just the first \(N\) indices. For \(t\in[-A,A]\) and any \(x,y\), \[ |F_N(x,t)-F_N(y,t)| \leq\sum_{j\leq N}l_j|\lambda_j(x)-\lambda_j(y)|. \tag{35}\] This follows from the elementary difference bound for finite minima, integrated over the intervals on which a cutoff can occur. Changing the upper endpoint costs at most its change, since \(0\leq L_N\leq1\). It follows that \(U_N\) has gradient bounded by \(|\nabla b|+P\) locally in \(\Omega\). On \(V_j\), once \(N\geq j\), its endpoint can be fixed at \(\beta_j\), and its gradient is bounded by \(P\).

Bounded pointwise convergence of the finite masks gives local strong \(L^2\) convergence. Weak Sobolev compactness therefore proves that \(U\) is locally Sobolev on \[O=\Omega\cup\bigcup_jV_j,\] with the same gradient bounds. These yield the finite total energy estimate \[ \int_O|\nabla U|^2 \leq2\int_\Omega|\nabla b|^2+2\int_\mathbb CP^2<\infty. \tag{36}\] Here the open union is integrated once; no multiplicity is introduced by overlapping neighborhoods.

The residual set \(B_* = \widehat{\mathbb C}\setminus O\) is compact and bounded, and consists only of common point components. Every connected subset of \(B_*\) lies in one component of \(\widehat{\mathbb C}\setminus\Omega\), hence is a singleton. Thus \(B_*\) is totally disconnected. Moreover \(B_*\subset\partial\Omega\), and so it is conformally removable by inclusion. Theorem 10, applied to the continuous bounded function \(U\) and (36), proves \(U\in W^{1,2}(\widehat{\mathbb C})\). If \(B_*\) is empty, this conclusion already follows from the local Sobolev statement and the energy bound. We have established the conclusion for every mask system \(m\); denote its function and density by \(U_m,P_m\).

A typical line and compression. Choose a line \(\ell\) parallel to \(e\) on which all \(U_m\) are locally absolutely continuous and all \(P_m\) are locally integrable. Almost every such line has these properties, since there are only countably many systems. We also require \[ |\partial\Omega\cap\ell|=0, \qquad \sum_{j:\,B_{j,1}\cap\ell\ne\varnothing}l_j<\infty. \tag{37}\] The first condition holds on almost every line by Lemma 4 and Fubini. For the second, write \(\ell_\tau\) for the parallel line with transverse coordinate \(\tau\). Tonelli gives \[\int_\mathbb R\sum_{j:\,B_{j,1}\cap\ell_\tau\ne\varnothing}l_j\,\mathrm d\tau =2\sum_jl_jd_{j,1}<\infty,\] by Cauchy–Schwarz and (30)–(31). Fix a line satisfying all these conditions.

For \(N\geq1\) let \[I^{(N)}=\bigcup_{\substack{j>N\\B_{j,1}\cap\ell\ne\varnothing}}I_j, \qquad H_N(t)=\int_{-A}^{t}{\bf1}_{\mathbb R\setminus I^{(N)}}(s)\,\mathrm ds.\] Equation (37) implies \(|I^{(N)}|\to0\). For any \(x\in C\cap\ell\), the distance from \(x\) to each of the first \(N\) closed components is positive. Some system \(m\) therefore has an open line neighborhood \(J\) of \(x\) missing its first \(N\) balloons. On \(J\), every active cutoff belongs to the tail appearing in \(I^{(N)}\), and hence \[L_m(z,t)\geq{\bf1}_{\mathbb R\setminus I^{(N)}}(t).\] The fixed-endpoint variation estimate extends to the infinite mask: for any segment \([x,y]\subset\ell\) and fixed \(t\in[-A,A]\), \[ |F_m(x,t)-F_m(y,t)|\leq\int_{[x,y]}P_m\,\mathrm ds. \tag{38}\] Indeed (32) and (35) give this for finite masks, and dominated convergence in the integration variable gives the limit. The estimate uses only the globally Lipschitz cutoffs, so it remains valid when the segment crosses indexed components where \(b\) is discontinuous.

If \(x,y\in C\cap J\), compare the two upper endpoints at one fixed location, and then use (38). The nonnegative-integrand inequality above gives \[\begin{align*} |H_N(b(x))-H_N(b(y))| &\leq |U_m(x)-U_m(y)|+\int_{[x,y]}P_m\,\mathrm ds\\ &\leq\int_{[x,y]}(|U_m'|+P_m)\,\mathrm ds. \tag{39}\end{align*}\] For example, when \(b(x)\leq b(y)\), first bound the compressed increment by \(F_m(x,b(y))-F_m(x,b(x))\) and then replace \(F_m(x,b(y))\) by \(F_m(y,b(y))\). The other order is identical.

The set \(C\cap J\) has length zero. A general elementary consequence of (39) is therefore \[|H_N(b(C\cap J))|=0.\] To see this with outer length, work first in a compact subinterval of \(J\). Cover its null set of common points by an open set on which the integral of \(|U_m'|+P_m\) is arbitrarily small. On each component interval, (39) bounds the diameter of the image of the common points by that integral. Summing gives arbitrarily small interval covers of the image. A countable exhaustion proves the assertion on \(J\).

We recover the uncompressed values by the one-dimensional area formula. The common-point set is Borel, since it is the compact complement with the countable union of the indexed closed components removed. Its trace map is continuous in the relative topology. Thus \(b(C\cap J)\) is analytic and Lebesgue measurable. Put \(S=b(C\cap J)\setminus I^{(N)}\). The nondecreasing Lipschitz function \(H_N\) has derivative one almost everywhere outside \(I^{(N)}\), and hence \[|S|=\int_S|H_N'| =\int_\mathbb R\#\{t\in S:H_N(t)=u\}\,\mathrm du=0.\] The last integrand is supported on the null set \(H_N(b(C\cap J))\). This reasoning allows both noninjectivity of \(H_N\) and its null set of derivative exceptions.

For fixed \(N\), countably many of the neighborhoods \(J\) suffice to cover \(C\cap\ell\), by the countable base of the line. It follows that \(b(C\cap\ell)\setminus I^{(N)}\) is null. Finally \(|I^{(N)}|\to0\), so \(|b(C\cap\ell)|=0\). Since \(b(x)=\pi_e(f_*(x))\) on common points, this is the required conclusion. Empty or finite indexed lists are included by interpreting the corresponding sums and tails in the evident way. ◻

Proof of Theorem 1. If \(\Omega=\widehat{\mathbb C}\), its conformal image is also the sphere, by compactness, and its automorphisms are Möbius transformations. Assume henceforth that the complement is nonempty. Apply separate Möbius changes in source and target to move a domain point and its image to infinity. A further target affine change gives \[ f(z)=z+\sum_{n\ge1}\alpha_nz^{-n} \quad\text{near infinity}. \tag{40}\] These changes preserve circle domains and conformal removability. All complementary components are now bounded Euclidean disks or points. Their correspondence is supplied by Lemma 2.

Index by \(j\) precisely those corresponding pairs for which at least one component is nondegenerate. Write \[K_j=\overline D(a_j,r_j),\qquad K'_j=\overline D(b_j,s_j),\] allowing a zero radius to mean a singleton. The index set is finite or countable, and \[ \sum_jr_j^2<\infty,\qquad \sum_js_j^2<\infty,\qquad \sum_jr_js_j<\infty. \tag{41}\] Indeed the positive-radius disks on each side have disjoint interiors in a bounded region, and the last assertion follows by Cauchy–Schwarz. All other complementary components are common points: both the source and target component are singletons. Let \(\tau\) denote their correspondence.

Choose \(R\) strictly larger than a radius containing the source complement. Define a vector field \(F\) on \(\overline D_R\) by using \(f\) in \(\Omega\), the center \(b_j\) on \(K_j\), and \(\tau(p)\) at every common point \(p\). This is a bounded measurable field. Boundedness follows from Lemma 2, or from the bounded side of the image outer contour. Measurability follows since there are countably many indexed closed components, the common-point set is Borel, and its singleton trace map is continuous relative to that set. No continuity of \(F\) at an indexed component is asserted.