Growth Competitions on Spherically Symmetric Riemannian Manifolds

Abstract

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.

arXiv comment10 pages, 3 figures; v2: Added references. v3: Minor changes in introduction and section 2