  We extend to any simply connected K\"ahler manifold with non-positive
sectional curvature some conditions for interpolation in $\mathbb{C}$ and in
the unit disk given by Berndtsson, Ortega-Cerd\`a and Seip. The main tool is a
comparison theorem for the Hessian in K\"ahler geometry due to Greene, Wu and
Siu, Yau.
