Compactness of the normalized K-quasiconformal self-maps of the sphere

A normalized map fixes infinity and is a plane homeomorphism on the finite chart. In either of the two bounded stereographic source charts, [F1]–[F2] and $\mathrm{\parallel }Df{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\le K{J}_f$ give total weighted energy at most $4\pi K$: integrate the target weight $4\mathrm{/}(1+\mathrm{\mid }f{\mathrm{\mid }}^2{)}^2$ against $J_f$. For a circle of source radius $r$ centered at x, let L(r) be its spherical image length. Almost every circle is AC by [F3], and Cauchy–Schwarz gives $L(r{)}^2\le 2\pi r{\int }_{\mathrm{\mid }z-x\mathrm{\mid }=r}4\mathrm{\parallel }Df{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\mathrm{/}(1+\mathrm{\mid }f{\mathrm{\mid }}^2{)}^{2}ds$. Integrating with dr/r between $\delta$ and $\sqrt{\delta }$ yields $\int L(r{)}^{2}dr\mathrm{/}r\le 8{\pi }^{2}K$. Hence one such circle has $L(r)\le 4\pi \sqrt{K\mathrm{/}\log (1\mathrm{/}\delta )}$, uniformly in f and the center x.

The three fixed points have positive minimum pairwise chordal distance c. At each x at least two of them stay a fixed positive distance from x; a uniformly small source disk avoids those two. Use the short circle from step 1.1 surrounding the smaller disk of radius delta. Its image lies in a spherical cap of diameter at most twice its length. For sufficiently small delta this cap cannot contain both avoided fixed points. The complement of the cap is connected, so Jordan separation makes one image complementary component lie inside the cap and the other contain its exterior. The image of the source disk cannot be the latter component, because it would contain at least one of the two fixed points which the source disk avoids. Thus its diameter tends uniformly to zero. Euclidean and chordal distances are uniformly comparable on the two bounded source charts, proving common chordal equicontinuity. The inverse maps are normalized and have the same K by [F1], so the identical argument gives their equicontinuity.

Apply [F4] to a sequence and then its inverses on the obtained subsequence. We obtain uniform limits f and g. Uniform convergence and continuity show $g(f(z))=z$ and $f(g(z))=z$, so f is a homeomorphism with inverse g and fixes the three points. It preserves orientation: uniformly close sphere maps are homotopic by normalized straight-line interpolation of their unit-sphere values; homotopy invariance of the sphere degree preserves degree one, and for a homeomorphism this is the positive local orientation sign (The singular chain homotopy formula, Global sphere degree is the sum of local degrees).

We prove closure with the exact K, independently of circular-dilatation or quadrilateral-modulus continuity. On any compact source patch choose a target chart avoiding its compact f-image complement point. Uniform convergence gives bounded finite coordinate values for f_n there for large n. The area formula and distortion bound give a uniform local $L^2$ derivative bound. By [F5], pass to weak $L^2$ derivative limits on a smaller patch; uniform convergence and integration against test functions identify them as Df. Write $f_n=u_n+i{v}_n$. The distributional identity $J_{f_n}={\mathrm{\partial }}_x(u_n{\mathrm{\partial }}_y{v}_n)-{\mathrm{\partial }}_y(u_n{\mathrm{\partial }}_x{v}_n)$ follows by smooth approximation and commutation of mixed weak derivatives. Uniform convergence of u_n and weak $L^2$ convergence of the derivatives of v_n show that these Jacobians converge distributionally to $J_f$. For every nonnegative smooth compactly supported phi, weak lower semicontinuity and the bound on f_n give $\int \varphi \mathrm{\parallel }Df{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\le \mathrm{lim\hspace{0.17em}inf}\int \varphi \mathrm{\parallel }D{f}_n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\le K\lim \int \varphi J_{f_n}=K\int \varphi J_f$. Hence $\mathrm{\parallel }Df{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\le K{J}_f$ almost everywhere. If J_f is zero this forces Df zero; otherwise the singular-value ratio is at most K, equivalently the analytic Beltrami bound. The general ACL characterization applies to the resulting W1,2 class, and continuity identifies f pointwise with its ACL representative on almost every line. Thus f is analytically K-QC by its definition and geometrically K-QC by [F1], proving closure.

The subsequential limit is therefore in the normalized family. Sequential compactness and the supremum chordal metric give compactness by [F4]. For general uniformly convergent quasiconformal homeomorphisms with a homeomorphic limit, if $L=\mathrm{lim\hspace{0.17em}inf}{K}_{f_n}<\mathrm{\infty }$, take a subsequence whose constants tend to L. The local derivative argument in step 4.1 uses the uniformly bounded constants and passes their limit to give $\mathrm{\parallel }Df{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\le L{J}_f$. It follows that $K_f\le L$; if L is infinite the inequality is automatic. Orientation is preserved by the same homotopy argument. This proves the full lower-semicontinuity claim without a normalization assumption on that sequence.