Let \(\mathcal S\) be the normalized univalent maps on \(\Delta\). Prove that the maps in \(\mathcal S\) whose images are polygons are dense in \(\mathcal S\) for uniform convergence on compact subsets of \(\Delta\).
Solution
Fix \(f\in\mathcal S\). We first shrink its image slightly, and then approximate each resulting analytic Jordan domain from inside by polygonal domains. The normalized Riemann maps of those polygons converge back to the shrunken map.
1. For \(0<r<1\), the map\[ F_r(z)=\frac{f(rz)}r \]lies in \(\mathcal S\), has Jordan-domain image, and satisfies\[ F_r\longrightarrow f \]uniformly on compact subsets as \(r\to1^-\).
The normalization is immediate: \[ F_r(0)=0, \qquad F_r'(0)=1. \] Since \(f\) is univalent, so is \(F_r\). Moreover \(F_r\) extends univalently to the larger disk \(|z|<1/r\), so its restriction to \(S^1\) is an analytic Jordan curve. Hence \[ U_r:=F_r(\Delta) \] is a Jordan domain.
If \(K\Subset\Delta\), then \(rz\to z\) uniformly for \(z\in K\), and \(f\) is uniformly continuous on a slightly larger compact disk. Therefore \[ \sup_{z\in K}\left|\frac{f(rz)}r-f(z)\right|\longrightarrow0. \]
2. Let \(F\in\mathcal S\) and suppose \(U=F(\Delta)\) is a Jordan domain with analytic boundary. Then there are polygonal Jordan domains \(P_n\) such that\[ F(s_n\overline\Delta)\subset P_n\Subset U, \qquad s_n\to1^-. \]
Choose numbers \[ 0<s_n<t_n<1, \qquad s_n\to1. \] The curve \[ \Gamma_n=F(t_nS^1) \] is a real-analytic embedded circle, and the compact set \(F(s_n\overline\Delta)\) has positive distance from \(\Gamma_n\). It also has positive distance from \(\partial U\).
Join sufficiently many successive points \[ F(t_ne^{2\pi i k/N}) \] by straight line segments. Because \(\Gamma_n\) is a regular embedded \(C^1\) curve, for all sufficiently large \(N\) this polygonal interpolation is a simple closed curve contained in an arbitrarily small tubular neighborhood of \(\Gamma_n\). Taking the neighborhood small enough keeps the polygon inside \(U\) and disjoint from \(F(s_n\overline\Delta)\). The polygon is homotopic to \(\Gamma_n\) in that tubular neighborhood, so its winding number about every point of \(F(s_n\overline\Delta)\) is one. Its bounded complementary component \(P_n\) therefore satisfies \[ F(s_n\overline\Delta)\subset P_n\Subset U. \]
3. Let\[ \phi_n:(\Delta,0)\longrightarrow(P_n,0) \]be the Riemann map with\[ d_n:=\phi_n'(0)>0. \]Then\[ d_n\longrightarrow1 \qquad\text{and}\qquad \phi_n\longrightarrow F \]locally uniformly on \(\Delta\).
Since \(P_n\subset U\), the map \[ \psi_n=F^{-1}\circ\phi_n:\Delta\to\Delta \] fixes \(0\). Because \(F'(0)=1\), the Schwarz lemma gives \[ d_n=\psi_n'(0)\le1. \]
On the other hand, \(F(s_n\Delta)\subset P_n\), so \[ \chi_n(z)=\phi_n^{-1}(F(s_nz)) \] is a holomorphic self-map of \(\Delta\) fixing \(0\). Again by the Schwarz lemma, \[ \frac{s_n}{d_n}=\chi_n'(0)\le1. \] Thus \[ s_n\le d_n\le1, \] and hence \(d_n\to1\).
The family \((\psi_n)\) is normal. Any locally uniformly convergent subsequence has a limit \(\psi:\Delta\to\overline\Delta\) with \[ \psi(0)=0, \qquad \psi'(0)=\lim d_n=1. \] The Schwarz lemma equality case forces \(\psi(z)=z\). Hence every convergent subsequence has the same limit, and therefore \[ \psi_n\longrightarrow\operatorname{id}_\Delta \] locally uniformly. Consequently \[ \phi_n=F\circ\psi_n\longrightarrow F \] locally uniformly.
4. After rescaling, these Riemann maps belong to \(\mathcal S\) and still have polygonal images.
Set \[ g_n(z)=\frac{\phi_n(z)}{d_n}. \] Then \[ g_n(0)=0, \qquad g_n'(0)=1, \] so \(g_n\in\mathcal S\). Its image is the scaled polygon \[ g_n(\Delta)=d_n^{-1}P_n. \] Since \(d_n\to1\) and \(\phi_n\to F\) locally uniformly, \[ g_n\longrightarrow F \] locally uniformly.
5. Polygon-image maps are dense in all of \(\mathcal S\).
1. Then\[ F_j\to f \]locally uniformly. By <1>2–<1>4, for each \(j\) choose a polygon-image map \(g_j\in\mathcal S\) so close to \(F_j\) on\[ \{|z|\le1-1/j\} \]that\[ \sup_{|z|\le1-1/j}|g_j(z)-F_j(z)|<1/j. \]For every compact \(K\Subset\Delta\), this estimate applies for all sufficiently large \(j\), and hence\[ g_j\longrightarrow f \]uniformly on \(K\). This proves the required density.