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Register a new userThe Geometric Airy Curve Flow on R^n
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Chuu-Lian Terng
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Langer and Perline proved that if x is a solution of the geometric Airy curve flow on R^n then there exists a parallel normal frame along x(. ,t) for each t such that the corresponding principal curvatures satisfy the (n-1) component modified KdV (vmKdV_n). They also constructed higher order curve flows whose principal curvatures are solutions of the higher order flows in the vmKdV_n soliton hierarchy. In this paper, we write down a Poisson structure on the space of curves in R^n parametrized by the arc-length, show that the geometric Airy curve flow is Hamiltonian, write down a sequence of commuting Hamiltonians, and construct Backlund transformations and explicit soliton solutions.
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Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with the bilinear form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= sum_{i=1}^{2n+1} (-1)^{n+i-1} e_{i, 2n+2-i}$. A smooth $gamma: Rto R^{n+1,n}$ is {it isotropic} if $gamma, gamma_x, ldots, gamma_x^{(2n)}$ are linearly independent and the span of $gamma, ldots, gamma_x^{(n-1)}$ is isotropic. Given an isotropic curve, we show that there is a unique up to translation parameter such that $(gamma_x^{(n)}, gamma_x^{(n)})=1$ (we call such parameter the isotropic parameter) and there also exists a natural moving frame. In this paper, we consider two sequences of curve flows on the space of isotropic curves parametrized by isotropic parameter. We show that differential invariants of these isotropic curves satisfy Drinfeld-Sokolovs KdV type soliton hierarchies associated to the affine Kac-Moody algebra $hat B_n^{(1)}$ and $hat A_{2n}^{(2)}$ Then we use techniques from soliton theory to construct bi-Hamiltonian structure, conservation laws, Backlund transformations and permutability formulas for these curve flows.
We give the following results for Pinkalls central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund transformations and a permutability formula, (iii) infinitely many families of explicit solutions. We also solve the Cauchy problem for periodic initial data.
In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.
The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flow is integrable if the evolution equation of the local differential invariants of a solution of the curve flow is a soliton equation. For example, the Hodge star mean curvature flow on $R^3$ and on $R^{2,1}$, the geometric Airy flow on $R^n$, the Schrodingier flow on compact Hermitian symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a compact Lie group are integrable. In this paper, we give a survey of these results, describe a systematic method to construct integrable curve flows from Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the Cauchy problem of these curve flows.
As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo cite{Guo} proved that $n$-dimensional compact self-shrinkers in $mathbb{R}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $mathbb{S}^n(sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in $mathbb{R}^{n+1}$ with constant scalar curvature are isometric to the round sphere $mathbb{S}^n(sqrt{n})$(see also cite{Hui1}). Complete classifications of $n$-dimensional translating solitons in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature were given by Mart{i}n, Savas-Halilaj and Smoczykcite{MSS} and Ancari and Chengcite{AC}, respectively. In this paper we give complete classifications of $n$-dimensional complete self-shrinkers in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Mart{i}n, Savas-Halilaj and Smoczyk cite{MSS} and Ancari and Chengcite{AC}.
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