Conformal removability of compact sets

Suppose K satisfies the neighborhood-local condition, and let F be a sphere homeomorphism conformal off K. Postcompose by a Möbius map taking F(infinity) to infinity. The resulting sphere homeomorphism H fixes infinity and restricts to a finite plane homeomorphism conformal off K. Apply the local condition with U equal to the whole plane. Thus H is holomorphic everywhere there; it was already conformal near infinity because K is finite and compact. Injectivity gives its holomorphic inverse in each chart by [F2], so [F6] makes H and hence F Möbius. This proves local implies global.

Now assume K is globally removable and let h:U→C be a homeomorphic embedding conformal off K, with K compactly contained in open U. A positive distance delta from K to the complement of U exists. Different U components meeting K contain disjoint delta-balls centered on a bounded set, so only finitely many meet K. Their sets $E_j=K\cap U_j$ are compact: components are closed relative to U. By [F1], each E_j is globally removable. It suffices to show h conformal near each E_j; on U minus K it already is. In a fixed connected U_j choose a compact connected C containing E_j: cover it by finitely many small closed disks inside U_j and join their centers by finitely many polygonal paths in U_j using [F5]. Let r be small and choose finitely many centers on C whose r-balls cover C, with their number bounded by a constant times r to the power minus2 (choose one point per occupied grid cell). For $\rho (z)={\sum }_i\exp (-\mathrm{\mid }z-c_i{\mathrm{\mid }}^2\mathrm{/}{r}^2)$, these radii can be adjusted by a fixed factor so its minimum on C is at least exp(-1), whereas outside U_j its maximum tends to zero as r tends to zero: the centers remain a fixed positive distance from that complement and polynomially many exponentials have exponentially small tails. Choose by Sard a regular level c between these two bounds and take the component Omega0 of rho>c containing connected C. Its closure is compact in U_j. Its boundary is a compact regular real-analytic one-manifold; finitely many implicit graph charts give finitely many components, each an analytic Jordan curve. To see the last assertion, use the complete unique flow of its nonvanishing unit tangent field from [F5]: each orbit is open, the other orbits are open, so an orbit is a whole connected component. Without a period it would identify that compact component homeomorphically with the real line; therefore it is periodic and embedded. Thus Omega0 is a connected finitely bordered domain containing E_j, and h is holomorphic on collars of all its boundary curves.

Each complementary sphere component of Omega0 is a Jordan disk; likewise for h(Omega0), and boundary components correspond by h. This follows by Jordan separation: a connected finitely bordered region has one outer boundary and disjoint nonnested hole boundaries, so filling the complementary sides gives exactly those disks. Parameterize each source/target complementary disk conformally by the unit disc after a Möbius chart normalization, using [F2]. The parameter maps extend analytically with nonzero derivative over their circles. Indeed an analytic boundary arc has a holomorphic parametrization with nonzero derivative and local holomorphic inverse; flatten that target arc and the source circle, then apply Schwarz reflection. If the first nonzero boundary Taylor term had degree at least2, its image of the half-disc would meet both sides of the flattened boundary, contrary to the mapped Jordan side; hence its degree is1. Compactness then makes each induced circle boundary map of h an analytic bi-Lipschitz diffeomorphism, and therefore quasisymmetric. Extend it by F3, and conjugate by the two disk parameter maps. This gives a QC homeomorphism of each closed complementary disk agreeing with h on its boundary.

Paste these finitely many disk maps to h on the closure of Omega0. Their domains and images cover the sphere with matching boundaries, so this is a sphere homeomorphism g equal to h on Omega0. It is QC off E_j with one finite common bound: there are finitely many disk extensions; inside Omega0 minus E_j it is conformal. Across an analytic boundary, flatten a compact subarc by its holomorphic inverse coordinate and apply the earlier line-gluing result in [F3]. Conformal source/target chart changes preserve the bound. Finitely many subarcs cover the compact boundary curves, so no separate unproved analytic-curve gluing theorem is assumed.

On the sphere minus g(E_j), g inverse is locally QC with that common bound. Extend its Beltrami coefficient by zero on the compact g(E_j), obtaining a bounded measurable sphere coefficient. Choose its QC solution Phi by [F4]. The composition formula makes Phi composed with g conformal off E_j; global removability of E_j therefore makes it Möbius. On g(Omega0 minus E_j), the inverse of h is conformal, so the chosen coefficient is zero there; it is zero on g(E_j) by definition. Thus Phi satisfies the weak zero-Beltrami equation on all g(Omega0). Its already-global QC regularity and [F4]'s one-QC criterion make it holomorphic there with holomorphic inverse. Rearranging h as Phi inverse composed with the Möbius map shows h conformal on Omega0. Apply this to the finitely many E_j and combine with the given conformality outside K. This proves global implies neighborhood-local and the claimed equivalence under AC. The exact sphere existence and coefficient-equality assertion is supplied by the stable13 proof in [F4].