Cauchy integrals and Möbius geometry of curves (Q2454920)

This short paper contains six theorems concerning estimates and explicit representations of objects like the Szegö projection operator, the Hilbert-Schmidt norm of the Kerzman-Stein operator, and the Kerzman-Stein kernel. An example is the finding that the square of the absolute value of the Kerzman-Stein kernel in the variables \(z\) and \(w\) is exactly equal to \[ -\frac{1}{4\pi^2}\tan{\frac{\theta}{2}}d_{z}d_{w}\theta, \] where \(\theta\) is a kind of distance between \(z\) and \(w\) (called the ``Kerzman-Stein distance'' in the paper) which is not even necessarily positive. It is however invariant with respect to Möbius transformations. An introduction giving all the definitions and technicalities is also provided. The proofs are short and rather direct. The authors announce a longer and more detailed paper.