Self-similar solutions to the curve shortening flow

  title={Self-similar solutions to the curve shortening flow},
  author={Hoeskuldur Petur Halldorsson},
  journal={Transactions of the American Mathematical Society},
  • H. P. Halldorsson
  • Published 9 July 2010
  • Mathematics
  • Transactions of the American Mathematical Society
We give a classification of all self-similar solutions to the curve shortening flow in the plane. 
Self-shrinking Solutions to Mean Curvature Flow
Self-shrinking Solutions to Mean Curvature Flow
Classification of convex ancient free boundary curve shortening flows in the disc
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
In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affineExpand
Curve-Shortening Flow
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 itsExpand
The fundamental solutions of the curve shortening problem via the Schwarz function
Curve shortening in the z-plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy StSz = Szz , where S(z, t) is the SchwarzExpand
The area preserving curve shortening flow with Neumann free boundary conditions
Abstract We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve theExpand
Self-Similar Solutions to the Inverse Mean Curvature Flow in $$\mathbb {R}^2$$
In this paper, we obtain a complete list of all self-similar solutions of inverse mean curvature flow in $\mathbb{R}^2$.
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 fundamentalExpand
Soliton solutions to the curve shortening flow on the sphere
It is shown that a curve on the unit sphere is a soliton solution to the curve shortening flow if and only if its geodesic curvature is proportional to the inner product between its tangent vectorExpand
The Affine Shape of a Figure-Eight under the Curve Shortening Flow
We consider the curve shortening flow applied to a natural class of figureeight curves, those with dihedral symmetry and some monotonicity assumptions on the curvature and its derivatives. We proveExpand


Curvature evolution of plane curves with prescribed opening angle
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 sameExpand
Complete noncompact self-similar solutions of Gauss curvature flows I. Positive powers
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
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 ZΛgradientExpand
Two‐Dimensional Motion of Idealized Grain Boundaries
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 isExpand
Regularity Theory for Mean Curvature Flow
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 SingularExpand
Nonlinear Oscillations Dynamical Systems, and Bifurcations of Vector Fields
Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.-Expand
Applied Nonlinear Control
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 automotiveExpand
Singularities of the curve shrinking flow for space curves
Slotine & W
  • Li, Applied Nonlinear Control. Prentice Hall, New Jersey,
  • 1991