A note on the Alexander theorem on the complex plane (Q2903924)

This paper in one-variable complex analysis and Banach manifolds uses global analysis (such as Smale's form of the Sard theorem) and complex analytic tools (such as the Cauchy transform on curves and on discs) to provide an extension to an ingredient of a theorem of \textit{H. Alexander} [Invent. Math. 125, No. 1, 135--148 (1996; Zbl 0853.32003)]. NEWLINENEWLINENEWLINENEWLINE Theorem~1 contains a result in several parts about a map between Banach manifolds of loops in \({\mathbb C}\), while the rest of the paper is devoted to its proof.NEWLINENEWLINELet \(L\subseteq {\mathbb C}\) be a loop regarded as a connected embedded smooth manifold of class \(C^\infty\) in the complex plane \({\mathbb C}\), parametrized by and diffeomorphic to the unit circle \({\mathbb T}\). Denote by \(\alpha_n:{\mathbb T}\to L\) a loop that traces \(L\) out \(n\)-times in the positive sense for \(n\in{\mathbb Z}\). Let \(1\in{\mathbb T}\) be a basepoint of the circle and \(p\in L\) that of the loop \(L\). Consider the function spaces \(G_r\), \(F\), \(Z(f_0)\) and the map \(\Delta: F\to G_r\) given as follows. Let \(G_r=C^r(\overline{\mathbb D})\) be the Hölder class on the unit disc \(\mathbb D\) of a nonintegral order \(r>1\), \(F\subset G_{r+1}\) its subspace of all \(f\) that map the pointed circle \(({\mathbb T},1)\) to the pointed loop \((L,p)\), \(Z(f_0)\) for any \(f_0\in F\) the subspace of \(G_{r+1}\) of all \(f\) with \(f(1)=0\) and \(f(\zeta)\) in the tangent line to \(L\) with footpoint \(f_0(\zeta)\) (i.e., \(Z(f_0)\) is the set of all (first) variations of functions \(f\in F\) at \(f_0\in F\)), and \(\Delta(f)=\partial f/\partial\overline\lambda\) with a Wirtinger derivative.NEWLINENEWLINETheorem~1. Then the following hold.NEWLINENEWLINE(1) The space \(F\) is a smooth (real) Banach manifold of class \(C^\infty\) and the mapping \(\Delta: F\to G_r\) is smooth of class \(C^\infty\). NEWLINENEWLINENEWLINENEWLINE (2) For \(f_0\in F\) the (real) tangent space \(T_{f_0}F\) can be indentified with \(Z(f_0)\), and so can the derivative \((d\Delta)(f_0)\) with the linear map \(Z(f_0)\ni f\mapsto{\partial f}/{\partial\overline\lambda}\in G_r\) featuring a Wirtinger derivative \(\partial/\partial\overline\lambda\). NEWLINENEWLINENEWLINENEWLINE (3) The connected components of \(F\) are \(S_n\), \(n\in{\mathbb Z}\), where \(S_n\) is the set of all discs \(f\in F\) whose boundary function \(f|{\mathbb T}\) is loop homotopic to \(\alpha_n\) through basepoint preserving loops \(({\mathbb T},1)\to(L,p)\).NEWLINENEWLINE(4) If \(n\in{\mathbb Z}\) and \(f_0\in F\), then the linear map \((d\Delta)(f_0)\) has (real) nullity \(\max\{n,0\}\) and corank \(\max\{-2n,0\}\). In particular, \(\Delta\) is a Fredholm mapping on each \(S_n\) of Fredholm index \(n+\min\{n,0\}\).NEWLINENEWLINEThe proofs call among other things upon some facts about the Cauchy transform such as that an area Cauchy transform on a disc is one degree more Hölder regular than its transformand, which requires the order \(r\) of the Hölder class to be nonintegral.