Grid Peeling and the Affine Curve-Shortening Flow

Grid peeling and the affine curve-shortening flow

In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call grid peeling) is the convex-layer decomposition of subsets G ⊂ Z of the integer grid, previously studied for the particular case G = {1, . . . ,m} by Har-Peled and Lidický (2013). The second...

Curve Shortening Flow in a Riemannian Manifold

In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let M be a compact locally symmetric space. If the curve shortening flow exists for...

An Introduction to Curve-Shortening and the Ricci Flow

Peeling the Grid

Consider the set of points formed by the integer n × n grid, and the process that in each iteration removes from the point set the vertices of its convex-hull. Here, we prove that the number of iterations of this process is O ( n4/3 ) ; that is, the number of convex layers of the n× n grid is Θ ( n4/3 ) . ∗Department of Computer Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL...

Blow-up rates for the general curve shortening flow

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in R2.