Growth Competitions on Spherically Symmetric Riemannian Manifolds

TL;DR

This paper models growth competition on spherically symmetric Riemannian manifolds, revealing how curvature influences whether competing sets remain bounded or grow indefinitely, with implications for geometric growth processes.

Contribution

It introduces a novel growth competition model on Riemannian manifolds and analyzes how surface curvature affects the long-term behavior of the competing sets.

Findings

On conformally Euclidean surfaces, the slower set remains bounded.

On hyperbolic surfaces, both sets can grow indefinitely.

Surface curvature determines the growth dynamics of competing sets.

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.