共查询到20条相似文献,搜索用时 31 毫秒
1.
Tai-Chia Lin Chi-Cheung Poon Dong-Ho Tsai 《Calculus of Variations and Partial Differential Equations》2009,34(2):153-178
We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to
a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also
have estimate on the curvature blow-up rate. 相似文献
2.
Glen Wheeler 《Annali di Matematica Pura ed Applicata》2013,192(5):931-950
In this paper, we consider the steepest descent H ?1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised L 2 oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large ‘waiting time’, the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one. 相似文献
3.
This paper deals with a non-local evolution problem for closed convex plane curves which preserves the perimeter of the evolving
curve but enlarges the area it bounds and makes the evolving curve more and more circular during the evolution process. And
the final shape of the evolving curve will be a circle in the C
∞ metric as the time t goes to infinity.
The first author is supported in part by the National Science Foundation of China (No.10671066) and the Shanghai Leading Academic
Discipline Project (No. B407). 相似文献
4.
Convexity and the Average Curvature of Plane Curves 总被引:1,自引:0,他引:1
The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length. If a rectifiable closed curve in R2 is contained in the interior of a convex set D then its average curvature is at least as large as the average curvature of the simple closed curve D which bounds the convex set. 相似文献
5.
Contraction of convex hypersurfaces in Euclidean space 总被引:5,自引:0,他引:5
Ben Andrews 《Calculus of Variations and Partial Differential Equations》1994,2(2):151-171
We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached. In the particular case of the mean curvature flow this provides a simple new proof of a theorem of Huisken.This work was carried out while the author was supported by an Australian Postgraduate Research Award and an ANUTECH scholarship. 相似文献
6.
Esther Cabezas-Rivas Carlo Sinestrari 《Calculus of Variations and Partial Differential Equations》2010,38(3-4):441-469
We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere. 相似文献
7.
8.
《Journal of Functional Analysis》2023,284(10):109904
In this paper, we investigate the long-term behavior for an invariant plane curve flow, whose evolution process can be expressed as a second-order nonlinear parabolic equation with respect to centro-affine curvature. The forward and backward limits in time are discussed, which shows that a closed convex embedded curve may converge to an ellipse when evolving according to this flow. In addition, we obtain the isoperimetric inequality in centro-affine plane geometry. 相似文献
9.
We prove some basic properties of Donaldson’s flow of surfaces in a hyperkähler 4-manifold. When the initial submanifold is symplectic with respect to one Kähler form and Lagrangian with respect to another, we show that certain kinds of singularities cannot form, and we prove a convergence result under a condition related to one considered by M.-T. Wang for the mean curvature flow. 相似文献
10.
Oliver C. Schnürer Knut Smoczyk 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2003,20(6):3302-1073
We consider the flow of a strictly convex hypersurface driven by the Gauß curvature. For the Neumann boundary value problem and for the second boundary value problem we show that such a flow exists for all times and converges eventually to a solution of the prescribed Gauß curvature equation. We also discuss oblique boundary value problems and flows for Hessian equations. 相似文献
11.
Mitsunori Nara 《Journal of Differential Equations》2007,237(1):61-76
The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane. 相似文献
12.
In this paper, we consider a new length preserving curve flow for closed convex curves in the plane. We show that the flow exists globally, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C ?? topology as t ?? ??. 相似文献
13.
We consider a motion of non-closed planar curves with infinite length. The motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature. We call the flow shortening–straightening flow. In this paper, first we prove a long time existence result for the shortening–straightening flow for non-closed planar curves with infinite length. Then we show that the solution converges to a stationary solution as time goes to infinity. Moreover we give a classification of the stationary solution. 相似文献
14.
Hermann Karcher 《manuscripta mathematica》1970,2(1):77-102
A generalized version of Alexandrow's angle comparison theorems is stated (1) and the following applications are given: A new proof of Klingenberg's estimate of the cut locus distance and related equality discussions (2), an existence proof for geodesic loops and for one closed geodesic (3), a new proof for the convexity of metric balls (4), a lemma concerning approximation of convex curves by polygons (5), lower and (known) upper bounds for the length of convex curves in terms of their geodesic curvature and the Gaussian curvature (6) and another comparison theorem for geodesic triangels (7). 相似文献
15.
We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive
sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic
or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions.
In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the
Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic.
Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one
containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification
of finite volume quotients of the complex hyperbolic space.
Oblatum 2-IX-1994 & 7-VIII-1995 相似文献
16.
It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and
a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time. 相似文献
17.
Philip Broadbridge Joanna M. Goard 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(3):534-538
An exact solution is given for the evolution of an
initially v-shaped surface by a fully nonlinear diffusion
equation. This is the unique generalized solution that is
continuous but not twice differentiable. Since the profile
velocity decreases faster than the reciprocal of the profile
curvature, the point of infinite curvature persists for a finite
positive time. 相似文献
18.
We study the evolution of a closed immersed hypersurface whose speed is given by a function f(H){\phi(H)} (H) of the mean curvature asymptotic to H/ ln H for large H. Compared with other nonlinear functions of the curvatures, this speed has some good properties which allow for an easier
study of the formation of singularities in the nonconvex case. We prove apriori estimates showing that any surface with positive
mean curvature at the initial time becomes asymptotically convex near a singularity. Similar estimates also hold for the mean
curvature flow; for the flow considered here they admit a simpler proof based only on the maximum principle. 相似文献
19.
In this paper we shall discuss a weighted curvature flow for a regular curve in the 2D Euclidean space. The weighted curvature flow for planar curves is a generalization of the well-known curvature flow discussed by Gage, Hamilton and Grayson. Under a suitable weighted curvature flow, convex curves will remain convex in the deformation process. However, the curve may not converge to a round point for general weights. Indeed, for a nonnegative weight function ω(u) with k isolated zeros, a curve will converge to a limiting k-polygon. The weighted curvature flow will have many useful properties which have applications to image processing. We shall also present some numerical simulations to illustrate how curves deform under the weighted curvature flow with different weight functions ω(u). Moreover, our algorithm is very effective and stable. The approximation of higher derivatives in our new algorithm only involve in the neighboring points. 相似文献
20.
Steven J. Altschuler Lang F. Wu 《Calculus of Variations and Partial Differential Equations》1994,2(1):101-111
In this work, we study surfaces over convex regions in 2 which are evolving by the mean curvature flow. Here, we specify the angle of contact of the surface to the boundary cylinder. We prove that solutions converge to ones moving only by translation.Partially supported by the NSF grant no: DMS-9100383Partially supported by the NSF grant no: DMS 9108269.A01 相似文献