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}
}
  • J. Ai, K. Chou, Juncheng Wei
  • Published 1 November 2001
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
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. 
View on Springer
59 Citations
Highly Influential Citations
9
Background Citations
20
Methods Citations
7
Results Citations
3

59 Citations

Citation Type

Citation Type

2π-periodic self-similar solutions for the anisotropic affine curve shortening problem
AbstractWe study the existence of 2π-periodic positive solutions of the equation $$u_{\theta\theta}+u=\displaystyle{\frac{a(\theta)}{u^3}},$$where a(θ) is a positive smooth 2π-periodic function. A…
Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow
  • Jian Lu
  • Mathematics
    International Mathematics Research Notices
  • 2018
In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R}…
EXTREMAL METRICS FOR CONFORMAL CURVATURES ON S 1
We define two conformal structures on S which give rise to a different view of the affine curvature flow and a new curvature flow, the “Qcurvature flow”. The steady state of these flows are studied.…
$2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II
The existence of $2\pi$-periodic positive solutions of the equation $$ u'' + u = \displaystyle{\frac{a(x)}{u^3}} $$ is studied, where $a$ is a positive smooth $2\pi$-periodic function. Under…
Steady States for One Dimensional Conformal Metric Flows
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…
STEADY STATES FOR ONE-DIMENSIONAL CURVATURE FLOWS
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…
Exact solutions of nonlinear conformally invariant integral equations in R 3
A UNIFIED FLOW APPROACH TO SMOOTH, EVEN
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…
A priori estimates and existence of solutions to the prescribed centroaffine curvature problem
A unified flow approach to smooth, even Lp-Minkowski problems
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…
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 19 REFERENCES
The scalar curvature equation on 2- and 3-spheres
In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this…
Remarks on prescribing Gauss curvature
We study the nonlinear partial differential equation for the problem of prescribing Gauss curvature K on S 2 . We give an example of a rotationally symmetric K for which the Kazdan-Warner obstruction…
Evolving convex curves
Abstract. We consider the behaviour of convex curves undergoing curvature-driven motion. In particular we describe the long-term behaviour of solutions and properties of limiting shapes, and prove…
On Conformal Deformations of Metrics onSn
Abstract On S n , there is a naturally metric defined n th order conformal invariant operator P n . Associated with this operator is a so-called Q -curvature quantity. When two metrics are pointwise…
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 ZΛgradient…
On affine plane curve evolution
Abstract An affine invariant curve evolution process is presented in this work. The evolution studied is the affine analogue of the Euclidean Curve Shortening flow. Evolution equations, for both…
Anisotropic flows for convex plane curves
where V and k are, respectively, the normal velocity and curvature of the interface. Equation (1) is sometimes called the “curve shortening problem” because it is the negative L2-gradient flow of the…
The heat equation shrinking convex plane curves
Soient M et M' des varietes de Riemann et F:M→M' une application reguliere. Si M est une courbe convexe plongee dans le plan R 2 , l'equation de la chaleur contracte M a un point
A priori estimates for prescribing scalar curvature equations
We obtain a priori estimates for solutions to the prescribing scalar curvature equation (1) R(x)n-2 on Sn for n > 3. There have been a series of results in this respect. To obtain a priori estimates…
Evolving Plane Curves by Curvature in Relative Geometries
In (0.1) X :S × [0, ω) → IR is the position vector of a family of closed convex plane curves, kN is the curvature vector, with k being the curvature and N the inward pointing normal given by N =…
...
1
2
...