  We propose a model for a growth competition between two subsets of a
Riemannian manifold. The sets grow at two different rates, avoiding each other.
It is shown that if the competition takes place on a surface which is
rotationally symmetric about the starting point of the slower set, then if the
surface is conformally equivalent to the Euclidean plane, the slower set
remains in a bounded region, while if the surface is nonpositively curved and
conformally equivalent to the hyperbolic plane, both sets may keep growing
indefinitely.
