Global geometry under isotropic Brownian flows

TL;DR

This paper studies how a codimension one manifold's global geometric properties, specifically Lipschitz-Killing curvatures, evolve under an isotropic, volume-preserving Brownian flow, revealing new insights into flow growth behaviors.

Contribution

It provides explicit expressions for the expected growth rates of intrinsic volumes of manifolds under isotropic Brownian flows, focusing on global geometric evolution.

Findings

Derived formulas for expected Lipschitz-Killing curvature growth

Revealed global growth patterns differ from local flow analyses

Enhanced understanding of geometric flow dynamics

Abstract

We consider global geometric properties of a codimension one manifold embedded in Euclidean space, as it evolves under an isotropic and volume preserving Brownian flow of diffeomorphisms. In particular, we obtain expressions describing the expected rate of growth of the Lipschitz-Killing curvatures, or intrinsic volumes, of the manifold under the flow. These results shed new light on some of the intriguing growth properties of flows from a global perspective, rather than the local perspective, on which there is a much larger literature.