Monotonicity of Spatial Critical Points Evolving Under Curvature-Driven Flows

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 Sect. 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.

Appendix

First compute the three curves $\beta ^G_1(\alpha ), \beta ^G_2(\alpha ), \beta _3(\alpha )$ in Fig. 4. We use the notations of Fig. 6. The curves $\beta ^G_1(\alpha ), \beta ^G_2(\alpha )$ separate the domains with $N=10$ and $N=6$ critical points. We can compute these lines based on the conditions that the centre of gravity $G$ should coincide with $P_1,P_2$, respectively. In the case of $G\equiv P_1$ we have to write moment balance to the horizontal line passing through $P_1$:

$$\begin{aligned} \frac{h^2}{12}=\frac{a^2}{2} \end{aligned}$$

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and by substituting $h=0.5\tan (\alpha )$, $a=\tan (\beta )$ we get

$$\begin{aligned} \beta ^G_1(\alpha )=\arctan \left( \frac{\tan (\alpha )}{2\sqrt{3}}\right) . \end{aligned}$$

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Fig. 6

Geometry of the symmetric 5-gon

In the case of $G\equiv P_2$ we have

$$\begin{aligned} \frac{h}{2}\left( \frac{h}{3}+c\right) =a\left( \frac{a}{2}-c\right) \end{aligned}$$

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and by substituting $h=0.5\tan (\alpha )$, $a=\tan (\beta ), c=1/(2\tan (\alpha ))$ we get

$$\begin{aligned} \frac{\tan (\alpha )}{4}\left( \frac{\tan (\alpha )}{6}+\frac{1}{2\tan (\alpha )}\right) =\tan (\beta )\left( \frac{\tan (\beta )}{2}-\frac{1}{2\tan (\alpha )}\right) \end{aligned}$$

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which yields

$$\begin{aligned} \beta ^G_2(\alpha )=\arctan \left( \frac{1}{2\tan (\alpha )}\left( 1+\sqrt{1 +\frac{\tan ^4(\alpha )}{3}+\tan ^2(\alpha )} \right) \right) \end{aligned}$$

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For the invariant subspace we consider the geometry of the 5-gon where all edges are tangent to the largest inscribed circle and we can write

$$\begin{aligned} a-\frac{1}{2}=\frac{1}{2}\tan \left( \frac{\pi }{4}-\frac{\alpha }{2}\right) \end{aligned}$$

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yielding

$$\begin{aligned} \beta _3(\alpha )=\arctan \left( \frac{1}{2}\tan \left( \frac{\pi }{4}-\frac{\alpha }{2}\right) +\frac{1}{2}\right) . \end{aligned}$$

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In the case of fixed reference point we choose the centre of the largest inscribed circle which is the ultimate point under Eikonal abrasion. To find the critical curves $\beta ^{U}_i(\alpha ),(i=1,2)$ we write the conditions for $U\equiv P_i$, yielding

$$\begin{aligned} \beta ^{U}_1(\alpha )&= \arctan \left( \frac{\sin (\alpha )}{2}\right) \end{aligned}$$

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$$\begin{aligned} \beta ^{U}_2(\alpha )&= \arctan \left( \frac{1}{2}\frac{\tan (\alpha )+1}{\tan (\alpha )}\right) . \end{aligned}$$

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