The curve shortening flow with density of a spherical curve in codimension two

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...

Ergodicity of Stochastic Curve Shortening Flow in the Plane

We study models of the motion by mean curvature of an (1+1) dimensional interface with random forcing. For the well-posedness of the models we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution equations in the variational framework of Krylov-Rozovskĭı, replacing the standard coercivity assumption by a Lyapunov type condition. Ergodicity is established for the ...

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...

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.

An Introduction to Curve-Shortening and the Ricci Flow