A smoothly bounded pseudoconvex domain \(D \subset \mathbb C^n\) is said to be of corank one if the Levi form of its boundary has no more than one degenerate eigenvalue. For a distance function \(d\) on \(D\) we denote by \(d^i\) the associated inner distance. The first main result of the article under review is a precise upper and lower bound for the distances of Bergman and Kobayashi and the inner distance for the Carathéodory distance on a corank one pseudoconvex domain \(D\). For this purpose the authors introduce optimal inner domains of comparison \(Q(\zeta, \delta)\), where \(\zeta \in D\) lies near \(\partial D\) and \(\delta >0\). Using these domains a pseudodistance \(\varrho\) is constructed and it is shown that any of the above-mentioned distances can be estimated from above and from below by a constant times \(\varrho\). A second main result gives a limit formula for the Fridman invariant on a corank one domain \(D\) as above.
The next result is about non-existence of biholomorphic maps between certain bounded pseudoconvex domains:
Let \(D_1,D_2\) be bounded domains in \(\mathbb C^n\). Let \(p^0 \in \partial D_1\) and \(q^0 \in \partial D_2\) be given, such that \(\partial D_1\) is strongly pseudoconvex near \(p^0\) and \(\partial D_2\) is smoothly bounded and of finite type near \(q^0\) and of corank one at \(q^0\). Then there cannot exist a biholomorphic map \(f:D_1 \to D_2\) for which \(q^0\) lies in the cluster set of \(f\) at \(p^0\).
Two more results treat the question of boundary continuity for Kobayashi isometries. For this purpose the authors compute explicitly the Kobayashi differential metric of the “egg domain” \(E_{2m}:=\{z \mid |z_1|^{2m} + |z_2|^2+\dots+|z_n|^2 <1\}\), where \(m \geq 1\) is an arbitrary integer. This is then applied to prove the following theorem:
Let \(D_1,D_2\) be bounded domains in \(\mathbb C^n\). Let \(p^0 \in \partial D_1\) and \(q^0 \in \partial D_2\) and assume the rank of the Levi form of \(\partial D_1\) is at least \(n-2\) near \(p^0\). Assume that \(\partial D_1\) is smooth and of finite type near \(p^0\) and \(\partial D_2\) is strongly pseudoconvex near \(q^0\). Then any \(C^1\)-isometry (in the Kobayashi metric) \(f:D_1 \to D_2\) for which \(q^0\) lies in the cluster set of \(f\) at \(p^0\) extends continuously to \(\overline {D_1}\) near \(p^0\).
Finally the following non-existence result for \(C^1\) Kobayashi isometries between perturbations of \(E_{2m}\) and certain domains is shown:
Let \(D_1,D_2\) be bounded domains in \(\mathbb C^n\). Let \(p^0 \in \partial D_1\) and \(q^0 \in \partial D_2\). Suppose that \(\partial D_2\) is \(C^2\)-strongly pseudoconvex near \(q^0\). Further assume that, in suitable local coordinates, \(\partial D_1\) is given near \(p^0\) by \[ 2 \operatorname{Re} z_n + |z_1|^{2m} + |z_2|^2+\cdots+|z_{n-1}|^2 + R(z, \bar z) <0 \] with an integer \(m>1\) and a remainder term \(R\) that goes to zero faster than at least one of the monomials of weight one. Then there cannot exist a \(C^1\)-isometry \(f:D_1 \to D_2\) for which \(q^0\) lies in the cluster set of \(f\) at \(p^0\).