On isometries of the Kobayashi and Carath\'{e}odory metrics

This article considers isometries of the Kobayashi and Carath'{e}od-ory metrics on domains in ${\mathbf{C}}^n$ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincar'{e}'s theorem about biholomorphic inequivalence of ${\mathbf{B}}^n$, the unit ball in ${\mathbf{C}}^n$ and ${\mathrm{\Delta }}^n$, the unit polydisc in ${\mathbf{C}}^n$ and then provide few examples which $\mathit{\text{suggest}}$ that ${\mathbf{B}}^n$ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries $f:D_1\to D_2$ to the closures under purely local assumptions on the boundaries. As an application, we show that there is no isometry between a strongly pseudoconvex domain in ${\mathbf{C}}^2$ and certain classes of weakly pseudoconvex finite type domains in ${\mathbf{C}}^2$.