Title:

Tracking erosion by integers: the evolution of critical points under curvature-driven flows.

Authors:

Gabor Domokos Budapest

Gary W. Gibbons Cambridge

Abstract:

The best known mathematical models for the abrasion of sedimentary particles are curvature-driven flows [1]: a special class of nonlinear partial differential equations defining the evolution of a surface $\Sigma$ by the speed $v$ in the direction of its surface normal, and $v$ is given as a function of the principal curvatures $\kappa, \lambda$ of $\Sigma$:

(1) v=v(\kappa, \lambda)

While locally defined, curvature-driven flows have startling global properties, e.g. they can shrink curves and surfaces to round points. These features made these flows powerful tools to prove topological theorems which ultimately led, via their generalizations by Hamilton to Perelmans's celebrated proof of the Poincar\'e conjecture. As a by-product of these great efforts, in 1987 Grayson [2] proved that if $\Sigma$ is given as a distance function from a fixed reference $O$ then the number $N(t)$ of spatial critical points (extrema of the distance) is decreasing monotonically under the planar $v=\kappa$ flow, also called the curve shortening flow. In 3D the statement is not true, however, even in 1995 Damon [3] noted that there was a general belief that critical points can not be created under the heat equation which can be regarded as a linearized version of the $v=\kappa +\lambda$ mean curvature flow. Damon also gave an example of creation of critical points under the heat equation.

In image processing it was known since the works of Koenderink [4] that the heat equation is a fundamental tool in understanding the blurring of images and there was a general intuition that in the blurring process critical points are 'seldom' created. This was formalized by Kuijper [5] who showed that under some probabilistic assumptions this is indeed the case.

Our goal is to generalize Grayson's Lemma to general 2D curvature-driven flows $v=v(\kappa)$, extend Kiujper's probabilistic appraoch from the heat equation to general 3D curvature-driven flows of type (1). All previous results are related to critical points with fixed reference $O$ and we also illustrate that if $O$ coincides with the center of gravity $G$ then the motion of the latter can be also modeled by added, symmetric random noise.

Our results indicate that the number $N(t)$ of static equilibrium points of sedimentary particles is decreasing stochastically under curvature-driven abrasion. This suggests a Markov process which shows remarkable agreement with field data measured on pebbles along rivers and in coastal areas.

References:

[1] F.J. Bloore, The Shape of Pebbles {\it Mathematical Geology} {\bf 9} (1977) 113-122.

[2] M. A. Grayson, The heat equation shrinks embedded plane curves to round points. {\it J. Diff Geom} {\bf 26} (1987) 285-3147

[3] J. Damon, Local Morse theory for solutions to the heat equation and Gaussian blurring, {\it J. Diff. Eq.} \bf 115, \rm (1995) 368-401.

[4] J.J. Koenderink, The structure of images. {\it Biol. Cybern.} {\bf 50} (1984) 363-370.

[5] A. Kuijper, The deep structure of Gaussian scale space {\it PhD Thesis} Chapter 7. (2002) University of Utrecht, Department of Mathematics.