# 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} }

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

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

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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

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

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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 −∞

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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

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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

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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 $\mathbb{R}^3$

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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

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