# Self-similar solutions to the curve shortening flow

@article{Halldorsson2012SelfsimilarST, title={Self-similar solutions to the curve shortening flow}, author={Hoeskuldur Petur Halldorsson}, journal={Transactions of the American Mathematical Society}, year={2012}, volume={364}, pages={5285-5309} }

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

## 83 Citations

### Ancient solutions to the Curve Shortening Flow spanning the halfplane

- MathematicsTransactions of the American Mathematical Society
- 2020

In this note we construct an infinite family of ancient solutions to the Curve Shortening Flow which span the halfplane.

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

- Mathematics
- 2015

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…

### Self-similar curve shortening flow in hyperbolic 2-space

- Mathematics
- 2021

We find and classify self-similar solutions of the curve shortening flow in standard hyperbolic 2-space. Together with earlier work of Halldórsson on curve shortening flow in the plane and Santos dos…

### Self-shrinking Solutions to Mean Curvature Flow

- Philosophy
- 2014

Self-shrinking Solutions to Mean Curvature Flow

### Classification of convex ancient free boundary curve shortening flows in the disc

- Mathematics
- 2021

We classify convex ancient curve shortening flows in the disc with free boundary on the circle.

### Affine self-similar solutions of the affine curve shortening flow I: The degenerate case

- MathematicsJournal of Differential Equations
- 2021

### Curve Shortening Flow on $\mathbb{T}^2$

- Mathematics
- 2022

. We present a characterization for the initial data of a soliton solution for the curve shortening ﬂow (CSF) on the torus of revolution T 2 ⊂ R 3 . Furthermore, we describe the behavior of such…

### Curve-Shortening Flow

- Mathematics
- 2020

Curve-shortening flow (CSF) is a geometric heat flow with a variety of applications in mathematics and physics that acts on each point of an immersed curve inwards at a speed proportional to its…

### A Compact Non-convex Ancient Curve Shortening Flow

- Mathematics
- 2022

. An example of a compact, non-convex, embedded ancient solution for the curve shortening ﬂow, which is asymptotic to Yin-Yang curve, is constructed.

### The Dirichlet problem for curve shortening flow

- Mathematics
- 2012

We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we…

## References

SHOWING 1-10 OF 10 REFERENCES

### Curvature evolution of plane curves with prescribed opening angle

- MathematicsBulletin of the Australian Mathematical Society
- 1995

We discuss the evolution of plane curves which are described by entire graphs with prescribed opening angle. We show that a solution converges to the unique self-similar solution with the same…

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

- Mathematics
- 1991

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…

### Complete noncompact self-similar solutions of Gauss curvature flows I. Positive powers

- Mathematics
- 1998

We classify all complete noncompact embedded convex hypersurfaces in $\mathbf{R}^{n+1}$ which move homothetically under flow by some negative power of their Gauss curvature.

### 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…

### Two‐Dimensional Motion of Idealized Grain Boundaries

- Mathematics
- 1956

To represent ideal grain boundary motion in two dimensions, a rule of motion of plane curves is considered whereby any given point of a curve moves toward its center of curvature with a speed that is…

### Regularity Theory for Mean Curvature Flow

- Mathematics
- 2003

1 Introduction.- 2 Special Solutions and Global Behaviour.- 3 Local Estimates via the Maximum Principle.- 4 Integral Estimates and Monotonicity Formulas.- 5 Regularity Theory at the First Singular…

### Applied Nonlinear Control

- Computer Science
- 1991

Covers in a progressive fashion a number of analysis tools and design techniques directly applicable to nonlinear control problems in high performance systems (in aerospace, robotics and automotive…

### Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

- MathematicsApplied Mathematical Sciences
- 1983

Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.-…

### E-mail address: hph@math

### Slotine & W

- Li, Applied Nonlinear Control. Prentice Hall, New Jersey,
- 1991