Ergodicity, bounded holomorphic functions and geometric structures in rigidity results on bounded symmetric domains

Over the years the author has been interested in rigidity problems on bounded symmetric domains of rank ≥ 2. In this article we give an overview on rigidity problems arising from holomorphic mappings either on bounded symmetric domains of rank ≥ 2 or on their finite-volume quotient manifolds into complex manifolds, placing the focus on recent developments. The article highlights the use of some fundamental elements in the theory, including ergodicity, bounded holomorphic functions and geometric structures. Especially, bounded holomorphic functions play an important role linking up with the other key elements of the theory. On the one hand, certain notions of extremal bounded holomorphic functions are essential for the proof of rigidity results arising from integral formulas on Chern forms and involving the use of Ergodic Theory. These results enlarge the scope of study of rigidity phenomena on holomorphic maps equivariant with respect to a lattice, allowing the target manifolds to be arbitrary bounded domains. On the other hand, integral representations of boundary values of bounded holomorphic functions, used in conjunction with Ergodic Theory, allow us to give a function-theoretic proof of the same results with strengthened applications. At the same time, the same tool in Harmonic Analysis allows us to recover proper holomorphic maps from admissible limits on boundary components, and the approach is now linked in rigidity problems with the study of geometric structures, more specifically with the geometric theory of varieties of minimal rational tangents (VMRTs) that the author has been developing with J.-M. Hwang in the study of uniruled projective manifolds in Algebraic Geometry.