On the monotonicity of spatial critical points evolving under curvature-driven flows

TL;DR

This paper investigates how the number of spatial critical points of smooth curves and surfaces changes under curvature-driven flows, showing monotonic behavior under certain conditions and extending previous results to stochastic settings.

Contribution

It proves monotonicity of critical points for curvature flows with specific speed dependencies and generalizes prior results to stochastic cases for surfaces in 3D.

Findings

Number of critical points decreases when speed derivative w.r.t. curvature is positive.

Number of critical points increases when speed derivative w.r.t. curvature is negative.

Results suggest natural processes tend to reduce static equilibrium points.

Abstract

We describe the variation of the number $N(t)$ of spatial critical points of smooth curves (defined as a scalar distance $r$ from a fixed origin $O$) evolving under curvature-driven flows. In the latter, the speed $v$ in the direction of the surface normal may only depend on the curvature $\kappa$. Under the assumption that only generic saddle-node bifurcations occur, we show that $N(t)$ will decrease if the partial derivative $v_{\kappa}$ is positive and increase if it is negative (Theorem 1). Justification for the genericity assumption is provided in Section 5. For surfaces embedded in 3D, the normal speed $v$ under curvature-driven flows may only depend on the principal curvatures $\kappa, \lambda$. Here we prove the weaker (stochastic) Theorem 2 under the additional assumption that third-order partial derivatives can be approximated by random variables with zero expected value and covariance. Theorem 2 is a generalization of a result by Kuijper and Florack for the heat equation. We formulate a Conjecture for the case when the reference point coincides with the centre of gravity and we motivate the Conjecture by intermediate results and an example. Since models for collisional abrasion are governed by partial differential equations with $v_{\kappa},v_{\lambda}>0$, our results suggest that the decrease of the number of static equilibrium points is characteristic of some natural processes.