General Mathematics, Vol. 5, No. 1 - 4, pp. 161-174, 1995
Abstract: Several families of complex geodesics of $D^{2}=\{z=(z_{1},z_{2})\in \C^{2}:|z_{1}|+|z_{2}|<1\}$ touching given boundary points are explicitly studied and the corresponding holomorphic retracts in the sense of Lempert [12] are founded. A careful investigation of the holomorphic extendibility of the Lempert--type retracts to the boundary of $D^{2}$ and a few results on their ``uniqueness'' allows the proof of a Cartan--type rigidity theorem at the boundary for holomorphic mappings of the domain $D^{2}$. Analogous rigidity results are obtained for special points of the boundary of $D^{n}$, $n>2$.
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