A Jordan Curve Theorem on a 3D Ball Through Brownian Motion

Abstract

The Jordan curve theorem states that any simple closed curve in 3D space divides the space into two regions, an interior and an exterior. In this article, we prove the Jordan curve theorem on the boundary of a 3D ball that is inserted in a complex plane bundle. To do so, we make use of the Brownian motion principle, which is a continuous-time and continuous-state stochastic process. We begin by selecting a random point on an arbitrarily chosen complex plane within a bundle G and on the boundary of the 3D ball considered. Using the two-step random process developed on complex planes earlier by Srinivasa Rao (Multilevel contours on bundles of complex planes, 2022), we draw a contour from the initial point to the next point on this plane. We then continue this process until we finish the Jordan curve that connects points on the boundary of a ball within G.