Contractions of invariant Finsler forms on the classical domains (Q1062184)

The author investigates nonconstant holomorphic maps on the classical symmetric Cartan domains D which contract all invariant Finsler forms. The main result of this paper is that these maps are biholomorphic. First one shows: If a linear map \(\Lambda\) : \(T_ 0(D)\to T_ 0(D)\) does not increase any invariant Finsler form at 0, then: (a) for each \(Y\in T_ 0(D)\) there exists a constant \(c\geq 0\) such that the canonical values of \(\Lambda\) (Y) are c times those of Y, (b) G is a multiple of a linear isometry on \(T_ 0(D)\) in the operator norm. - Next one shows: Suppose \(f: D\to D\) is a holomorphic map such that \(f(0)=0\) and Df(0)\(\neq 0\). If for each \(z\in D\) there exists a scalar \(a_ z\) and a linear isometry \(L_{(z)}\) such that \(Df_ z(0)=a_ zL_{(z)}\), then f is itself a linear isometry. From these results the main theorem follows.