  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.
