# Self-similar solutions for the anisotropic affine curve shortening problem

@article{Ai2001SelfsimilarSF, title={Self-similar solutions for the anisotropic affine curve shortening problem}, author={Jun Ai and Kai-Seng Chou and Juncheng Wei}, journal={Calculus of Variations and Partial Differential Equations}, year={2001}, volume={13}, pages={311-337} }

Abstract. Similarity between the roles of the group $SL(2,\bf R)$ on the equation for self-similar solutions of the anisotropic affine curve shortening problem and of the conformal group of $S^2$ on the Nirenberg problem for prescribed scalar curvature is explored. Sufficient conditions for the existence of affine self-similar curves are established.

## 59 Citations

2π-periodic self-similar solutions for the anisotropic affine curve shortening problem

- Mathematics
- 2011

AbstractWe study the existence of 2π-periodic positive solutions of the equation
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Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

- MathematicsInternational Mathematics Research Notices
- 2018

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}…

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$2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II

- Mathematics
- 2015

The existence of $2\pi$-periodic positive solutions of the
equation
$$
u'' + u = \displaystyle{\frac{a(x)}{u^3}}
$$
is studied, where $a$ is a positive smooth
$2\pi$-periodic function. Under…

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- 2006

We define two conformal structures on $S^1$ which give rise to a different view of the affine curvature flow and a new curvature flow, the ``$Q$-curvature flow". The steady state of these flows are…

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- Mathematics
- 2006

We define two conformal structures on S1 which give rise to a different view of the affine curvature flow and a new curvature flow, the "Q-curvature flow". The steady states of these flows are…

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- 2018

We study long-time existence and asymptotic behavior for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to…

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- 2018

A unified flow approach to smooth, even
Lp-Minkowski problems

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We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to…

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