Univalent functions and \(\mathrm{Diff}(S^1)/S^1\)
The author considers the space \(\mathcal A\) of univalent functions that are analytic in the unit disk (and apparently continuous and differentiable in the closed unit disk, but it is not stated explicitly). The paper contains two theorems. In the first theorem the author claims that there is a differentiable invertible map \(\phi_C\) from the space Diff\((S^1)/S^1\) of diffeomorphisms of the unit circle modulo rotations onto the space \(\mathcal A\). The map \(\phi_C\) is not defined in the paper. The reader is refered to two papers from the year 1987. Also, no differentiable structure is indicated on \(\mathcal A\) or on Diff\((S^1)/S^1\) . The inverse map is defined as the restriction of functions from \(\mathcal A\) to the unit circle \(S^1\), but the restriction does not take \(S^1\) to \(S^1\) in general, so the definition is not precise. The second theorem extends this result to the supersymmetric generalization \(S^{1,1}\) of \(S^1\) and to the space of superanalytic functions on \(S^{1,1}\).
scientific article; zbMATH DE number 6102817