  A recent paper [CGT] studies the evolution of star-shaped mean convex
hypersurfaces of the Euclidean space by a class of nonhomogeneous expanding
curvature flows. In the present paper we consider the same problem in the real,
complex and quaternionic hyperbolic spaces, investigating how the richer
geometry of the ambient space affects the evolution. In every case the initial
conditions are preserved and the long time existence of the flow is proven. The
geometry of the ambient space influences the asymptotic behaviour of the flow:
after a suitable rescaling the induced metric converges to a conformal multiple
of the standard Riemannian round metric of the sphere if the ambient manifold
is the real hyperbolic space, otherwise it converges to a conformal multiple of
the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in
every cases, we are able to construct infinitely many examples such that the
limit does not have constant scalar curvature.
