In [Ann. Math., II. Ser. 125, 105-152 (1987; Zbl 0616.53040)] the second author established a rigidity theorem for locally symmetric Hermitian manifolds of finite volume uniformized by an irreducible bounded symmetric domain of rank \(\geq 2\). This article is concerned with the situation when the domain manifold is assumed to be of rank 1. The main result is: “Let X and Y be complex hyperbolic space forms of complex dimension n and m respectively. Assume that X is compact and \(m\leq 2n-1\). Let f: \(X\to Y\) be a holomorphic immersion. Then, f is necessarily an isometric totally geodesic immersion.” (Here, a complex hyperbolic space form is defined as a quotient space \(B^ n/\Gamma\), where \(B^ n\) is the unit ball in \({\mathbb{C}}^ n\) equipped with the Poincaré metric of constant holomorphic sectional curvature -2 and \(\Gamma\) is a torsion- free discrete group of automorphisms of \(B^ n.)\)
For the proof the authors firstly consider the case when X is immersed in Y as a submanifold. The essential object of their investigations is the closed non-negative (1,1)-form \(\sigma\) on X defined by \(\sigma =- (1/n+1)Ric(X)-\omega | X,\) where \(\omega\) \(| X\) is the restriction of the Kähler form \(\omega\) of Y to X and Ric(X) denotes the Ricci form of the Kähler manifold (X,\(\omega\) \(| X)\). Using some foliation techniques they derive \(\sigma =0\), which is equivalent to the vanishing of the second fundamental form of X. The deduction of the general case from the special one is mainly due to the uniqueness of Kähler-Einstein metrics of negative scalar curvature on a compact Kähler manifold.