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

2021 ◽  
Vol 285 ◽  
pp. 686-713
Author(s):  
Chengjie Yu ◽  
Feifei Zhao
Keyword(s):  
Degenerate Case ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

A note on the Abresch—Langer conjecture

2014 ◽  
Vol 144 (2) ◽  
pp. 299-304 ◽  
Author(s):  
Kai-Seng Chou ◽  
Xiao-Liu Wang
Keyword(s):  
Saddle Point ◽  
The Self ◽  
Point Property ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

A saddle-point property of the self-similar solutions in the curve shortening flow was conjectured by Abresch and Langer and confirmed by Au. An improvement on Au's solution is presented.

Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

2018 ◽  
Vol 2020 (23) ◽  
pp. 9440-9470
Author(s):  
Jian Lu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Smooth Function ◽  
Existence Result ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

Abstract In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}$$is studied, where $f$ is a positive $2\pi $-periodic smooth function. By virtue of a new generalized Blaschke–Santaló inequality, we obtain a new existence result of solutions.

scholarly journals TRANSLATING SOLITONS FOR LAGRANGIAN MEAN CURVATURE FLOW IN COMPLEX EUCLIDEAN PLANE

2012 ◽  
Vol 23 (10) ◽  
pp. 1250101 ◽  
Author(s):  
ILDEFONSO CASTRO ◽  
ANA M. LERMA
Keyword(s):  
Mean Curvature ◽  
Euclidean Plane ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions ◽  
Hamiltonian Stationary ◽  
Translating Solitons

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones [Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom.84 (2010) 127–161] in dimension two. The simplest (non-trivial) example in our family is characterized as the only (non-totally geodesic) Hamiltonian stationary Lagrangian translating soliton for mean curvature flow in complex Euclidean plane.

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