Hermitian metric rigidity on compact foliated manifolds (Q1900102)

Some rigidity type results are obtained for a compact foliated manifold \((M, {\mathcal F})\) equipped with a leafwise holomorphic structure and a smooth leafwise Kähler metric \(g\). For example, if each leaf is uniformized by a fixed irreducible bounded symmetric domain \(\Omega\) of rank \(\geq 2\), \(h\) is a smooth leafwise Hermitian metric of non-positive curvature in the sense of Griffiths, then \(g\) and \(h\) are homothetic on \(\mu\)-a.e. leaf, \(\mu\) being a finite invariant transverse measure for \(\mathcal F\). Some applications of this result are provided.