A correspondence for isometric immersions into product spaces and its applications

We present a correspondence for isometric immersions that are graphs in Riemannian or semi-Riemannian warped product spaces. We use this correspondence to give several existence and non-existence theorems for hypersurfaces in Riemannian or Lorentzian spaces. In the case of surfaces, we obtain further applications regarding height estimates, harmonic representation of surfaces and the existence of holomorphic quadratic differentials in homogeneous and non-homogeneous spaces.