# The Zoo of Solitons for Curve Shortening in $\R^n$

@inproceedings{JAltschuler2012TheZO,
title={The Zoo of Solitons for Curve Shortening in \$\R^n\$},
author={Dylan J.Altschuler and Steven J.Altschuler and Sigurd B.Angenent and Lani F.Wu},
year={2012}
}
• Published 17 July 2012
• Mathematics
We provide a detailed description of solutions of Curve Shortening in Rn that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curve shortening on the unit sphere Sn−1. Expanding rotating solitons turn out to be asymptotic to generalized logarithmic spirals. In terms of asymptotic properties of their ends the…
18 Citations

## Figures and Tables from this paper

### Solitons of Curve Shortening Flow and Vortex Filament Equation

In this paper we explore the nature of self-similar solutions of the Curve Shortening Flow and the Vortex Filament Equation, also known as the Binormal Flow. We explore some of their fundamental

### O(m)×O(n)-invariant homothetic solitons for inverse mean curvature flow in Rm+n

• Mathematics
• 2019
The inverse mean curvature flow (IMCF) has been extensively studied not only as a type of geometric flows, but also for its applications to geometric inequalities. The focus is primarily on

### Translating Solitons of the Mean Curvature Flow in Arbitrary Codimension

This is a survey of the Ph.D thesis by the author. In the thesis we study the translating solitons of the mean curvature flow. Although many researchers study translating solitons in codimension one,

### Translating solitons in arbitrary codimension

We study the translating solitons of the mean curvature flow. Although many authors study translating solitons in codimension one, there are few references and examples for higher codimensional cases

### A note on self-similar solutions of the curve shortening flow

This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The

### Codimension bounds and rigidity of ancient mean curvature flows by the tangent flow at −∞

• Mathematics
Communications in Contemporary Mathematics
• 2021
Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, by adapting the work of Colding–Minicozzi [11], we prove codimension bounds for ancient mean

### Solitons of Discrete Curve Shortening

• Mathematics
• 2015
For a polygon $${x=(x_{j})_{j \in \mathbb{z}}}$$x=(xj)j∈z in $${\mathbb{R}^{n}}$$Rn we consider the polygon $${(T(x))_j=\left\{x_{j-1}+2x_j+x_{j+1}\right\}/4.}$$(T(x))j=xj-1+2xj+xj+1/4. This

### Nonconvex ancient solutions to Curve Shortening Flow

• Physics
• 2022
A BSTRACT . We construct an ancient solution to planar curve shortening. The so- lution is at all times compact and embedded. For t (cid:191) 0 it is approximated by the rotating Yin-Yang soliton,

### Stabilization technique applied to curve shortening flow in $\mathbb{R}^3$

For the curve shortening flow in R3 we derive a family of monotonicity formulae depending on parameter λ. For λ = 1 our formula coincides with the classical formula of Huisken, while for λ 6= 1 the

### Ancient solutions to mean curvature flow for isoparametric submanifolds

• Mathematics
Mathematische Annalen
• 2020
Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied in Liu and Terng (Duke Math J 147(1):157–179, 2009). In this paper, we will show that all these

## References

SHOWING 1-10 OF 13 REFERENCES

### The normalized curve shortening flow and homothetic solutions

• Mathematics
• 1986
The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient

### Asymptotic-behavior for singularities of the mean-curvature flow

is satisfied. Here H(p,ή is the mean curvature vector of the hypersurface Mt at F(/?, t). We saw in [7] that (1) is a quasilinear parabolic system with a smooth solution at least on some short time

### The Curve Shortening Problem

• Mathematics
• 2001
BASIC RESULTS Short Time Existence Facts from Parabolic Theory Evolution of Geometric Quantities INVARIANT SOLUTIONS FOR THE CURVE SHORTENING FLOW Travelling Waves Spirals The Support Function of a

### Soliton solutions for the mean curvature flow

• Mathematics
• 2000
We consider soliton solutions of the mean curvature flow, i.e., solutions which move under the mean curvature flow by a group of isometries of the ambient manifold. Several examples of solitons on

### Shortening space curves and flow through singularities

• Mathematics
• 1992
When a closed curve immersed in the plane evolves by its curvature vec- tor, singularities can form before the curve shrinks to a point. We show how to use the curvature flow on space curves to

### Singularities of the curve shrinking flow for space curves

Singularities for space curves evolving by the curve shrinking flow are studied. Asymptotic descriptions of regions of the curve where the curvature is comparable to the maximum of the curvature are

### Self-similar solutions to the curve shortening flow

We give a classification of all self-similar solutions to the curve shortening flow in the plane.

• 2001

### Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states

• Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston
• 1989