Compressions and contractions of Hermitian symmetric spaces
Let \(D=G/K\) be a bounded symmetric domain equipped with the Bergman metric \(d\). We say that function \(f:D\to D\) is a contraction for the metric \(d\) if for all \(p\) and \(q\) in \(D\) \[ d\bigl(f(p), f(q)\bigr)\leq d(p,q). \] We say that \(f:D\to D\) is a strict contraction if there is a \(K<1\) so that \[ d \bigl(f(p),f(q)\bigr)\leq Kd(p,q). \] Let \(G_\mathbb{C}\) be the complexification of \(G\) and consider the so-called group of compressions of \(D\) \[ \Gamma= \bigl\{a \in G_\mathbb{C},g(D) \subset D\bigr\}. \] It is known that \(\Gamma=G\exp(C)\) where \(C\) is a convex, closed, proper cone. Let \(C^0\) be the interior of the cone \(C\) and denote by \(\Gamma^0\) the set \(G\exp C^0\). In this paper the author proves that every element \(\gamma\in\Gamma\) is a contraction for the Bergman metric of \(D\). Moreover, every \(\gamma\in\Gamma^0\) is a strict contraction for the Bergman metric of \(D\).
scientific article; zbMATH DE number 1229578