Proper holomorphic self-maps of bounded domains with a weakly spherical boundary point (Q461410)

The authors consider the class of bounded domains \(D \subset \mathbb C^2\) with \textit{weakly spherical boundary points}. These are points \(x \in \partial D\) for which there exists a local system of coordinates, in which \( x \equiv (0,0)\) and the domain admits a defining equation of the form \[ \text{Re\,} w + |z|^{2p} + g(z, w) < 0\;, \] for an integer \(p \geq 2\) and a function \(g\) with \[ \lim_{z \to 0}\frac{g(z,w)}{z^{2p}} = 0. \] The point \((0, 0)\) is called \textit{tame} if the (real-valued) function \(g\) is non-negative on a neighbourhood of \((0,0)\) and \(g(z, w) = 0\) if and only if \((z, w) = (0,0)\). We recall that the notion of weakly spherical points was introduced by \textit{E. Barletta} and \textit{E. Bedford} [Indiana Univ. Math. J. 39, No. 2, 315--338 (1990; Zbl 0707.32005)], where they proved that these boundary points are precisely those that occur in the branch loci of the proper holomorphic maps from pseudoconvex domains with real analytic boundaries onto strongly pseudoconvex domains with \(\mathcal C^2\) boundary. The main result of this paper consists in the proof that if a bounded domain \(D \subset \mathbb C^2\) with \(\mathcal C^\infty\) boundary has a tame weakly spherical boundary point \(x\) of finite vanishing order, then any proper holomorphic self-map \(f: D \to D\) extends smoothly to an open neighbourhood \(\mathcal U\) of \(x\). Using this, the authors are also able to show that if a proper holomorphic self map \(f\) fixes \(x\), then it necessarily fixes all inner points of the form \((0,w) \in D\) and at those points one has \(\big|\det J(f)|_{(0,w)}\big| = 1\). This yields that \(f\) is an automorphism of \(D\) by the classical Carathéodory-Cartan Theorem.