Complex geodesics, their boundary regularity, and a Hardy-Littlewood-type lemma (Q2790820)

Let \(\Omega\) be a bounded domain in \(\mathbb C^n\) and let \(\mathbb D\) denote the open unit disc centered at \(0\in\mathbb C\). A holomorphic map \(f: \mathbb D\rightarrow \Omega\) is called a complex geodesic of \(\Omega\) if it is an isometry for the Kobayashi distances on \(\mathbb D\) and \(\Omega\). NEWLINENEWLINENEWLINENEWLINE A domain \(\Omega\) is said to be strictly convex if for any two points \(z_1, z_2\in \overline{\Omega}\) the open segment \(\{tz_1 + (1 - t)z_2 : 0 < t < 1\} \subset \Omega\). NEWLINENEWLINENEWLINENEWLINE The author gives an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to \(\partial \mathbb D\). NEWLINENEWLINENEWLINENEWLINE Let \(\Omega\) be a bounded convex domain in \(\mathbb C^n\), \( n \geq 2\), with \(C^1\)-smooth boundary. Let \(F:B^{n-1}(0; r) \rightarrow \mathbb R\) be a smooth function with \(F(0) = 0\) and \(DF(0) = 0\). We say that \(F\) supports \(\Omega\) from the outside if there exist constants \(R_0 \in (0, r)\) and \(s_0 > 0\) such that, for each \(w\in \partial\Omega\), there exists a unitary transformation \(U_w\) satisfying: NEWLINE\[NEWLINEU^w(H_w(\partial\Omega)) = \{v \in \mathbb C^n: v_n = 0\},NEWLINE\]NEWLINE and \( U_w(\nu_w) = (0, \ldots , 0, i)\), where \(\nu_w\) denotes the inward unit normal vector to \(\partial\Omega\) at \(w\), such that, denoting the \(\mathbb C\)-affine map \(v \mapsto U_w(v -w)\) by \(U^w\), we have NEWLINE\[NEWLINEU_w(\overline{\Omega}) \cap B^{n-1}(0; R_0) \times \big((-s_0, s_0) + i(-s_0, s_0)\big)NEWLINE\]NEWLINE NEWLINE\[NEWLINE\subset \big\{z = (z^\prime, z_n) \in B^{n-1}(0; R_0) \times {\big ((-s_0, s_0) + i(-s_0, s_0)\big)}: \mathrm{Im}(z_n)\geq F(z^\prime)\big\}.NEWLINE\]NEWLINENEWLINENEWLINESet \(\Psi_\alpha(x) = \exp(-1/x^\alpha)\) if \(x > 0\), and equal to \(0\) if \(x=0\).NEWLINENEWLINEThe main result of this article is the following theorem. NEWLINENEWLINENEWLINENEWLINE Theorem. Let \(\Omega\) be a bounded convex domain in \(\mathbb C^n\), \(n \geq 2\), with \(C^1\)-smooth boundary. Assume that \(\Omega\) is supported from the outside by \(F(z^\prime) := C\Psi_\alpha( \| z^\prime\|^\alpha)\) (writing \(z = (z^\prime, z_n)\) for each \(z \in \mathbb C^n\)) for some \(C > 0\) and some \(\alpha \in (0, 1)\). Then every complex geodesic of \(\Omega\) extends continuously up to \(\partial D\).NEWLINENEWLINEThe author also establishes a Hardy-Littlewood-type lemma.