Sextactic points on a simple closed curve

The self-linking number of a closed curve in R^n

We introduce the self-linking number of a smooth closed curve in R^n with respect to a 3-dimensional vector bundle over the curve, provided that some regularity conditions are satisfied. When n=3, this construction gives the classical self-linking number of a closed embedded curve with non-vanishing curvature. We also look at some interesting particular cases, which correspond to the osculating or the orthogonal vector bundle of the curve.

Approximations of periodic functions to R^n by curvatures of closed curves

We show that for any n real periodic functions f_1,..., f_n with the same period, such that f_i>0 for i<n, and a real number e >0, there is a closed curve in R^{n+1} with curvatures k_1, ..., k_n such that |k_i(t)-f_i(t)| < e for all i and t. This neither holds for closed curves in the hyperbolic space H^{n+1}, nor for parametric families of closed curves in R^{n+1}.

The length of the shortest closed geodesics on a positively curved manifold

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

The length of the shortest closed geodesics on a positively curved manifold

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

On Some Characterizations of Ruled Surface of a Closed Spacelike Curve with Timelike Binormal in Dual Lorentzian Space

In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed spacelike curve with timelike binormal in dual Lorentzian space.

The anisotropic polyharmonic curve flow for closed plane curves

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{U}$ (where $\mathcal{U}\subset\mathbb{R}^{2}$ is a convex, centrally symmetric domain) induces a second convex body, the isoperimetrix $\tilde{\mathcal{I}}$. This set is the unique convex set that miniminises the isoperimetric ratio (modulo homothetic rescaling) in the Minkowski …

On Some Characterizations of Ruled Surface of a Closed Spacelike Curve with a Spacelike Binormal in Dual Lorentzian Space

In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed spacelike curve with a spacelike binormal in dual Lorentzian space.

On the optimality of one integral inequality for closed curves in $\mathbb R^4$

The optimality of the integral inequality $\int\limits_γ\sqrt{k_1^2+k_2^2+k_3^2}ds>2π$ for closed curves with non-vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive curvatures in $\mathbb R^4$ satisfies the inequality $\int\limits_γ\sqrt{k_1^2+k_2^2+k_3^2}ds \geq 2\sqrt{5}π$.

Instability of Closed $p$-Elastic Curves in $\mathbb{S}^2$

For $p\in\mathbb{R}$, we show that non-circular closed $p$-elastic curves in $\mathbb{S}^2$ exist only when $p=2$, in which case they are classical elastic curves, or when $p\in(0,1)$. In the latter case, we prove that for every pair of relatively prime natural numbers $n$ and $m$ satisfying $m<2n<\sqrt{2}\,m$, there exists a closed spherical $p$-elastic curve with non-constant curvature which winds around a pole $n$ times and closes up in $m$ periods of its curvature. Further, we show that all closed spherical $p$-elastic curves for $p\in(0,1)$ are unstable as critical points of the $p$-elast…

关于闭渐近曲线的Nirenberg猜想的一个反例

A counterexample to Nirenberg's conjecture on closed asymptotic curves

本文构造了R³中一条含两条闭渐近曲线的光滑嵌入环带，为Nirenberg1962年的猜想提供了反例，其构造基于Lelieuvre公式与Riccati方程周期解。

Curve shortening and the topology of closed geodesics on surfaces

We study "flat knot types" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial.

Closed ideal planar curves

In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply-covered circle. Moreover, we show that curves in any homotopy class with initially small $L^3\lVert k_s\rVert_2^2$ enjoy a uniform length bound under the flow, yielding the convergence result in these cases.

The polyharmonic heat flow of closed plane curves

In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C^infty-topology to a simple circle. Our results yield a characterisation of the total amount of time during which the flow is not strictly convex, quantifying in a sense the failure of the maximum principle.

在R³中闭合曲线的法线计数

Counting normals to closed curves in $\mathbb{R}^3$

该研究证明了在三维空间中，闭合曲线至少存在一定数量的法线，利用Morse理论和自交性质得出相关结论。

凸闭平面曲线的两个非局部逆曲率流

Two nonlocal inverse curvature flows of convex closed plane curves

本文提出两类非局部逆曲率流，使闭凸曲线长度递减、面积递增，并在无奇点时光滑收敛至圆。

Knot types, homotopies and stability of closed elastic rods

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the m-fold covered circle to the n-fold covered circle for any m,n relatively prime. Each family contains exactly one self-intersecting curve, one elastic curv…

具有指定曲率的平面闭曲线

Planar closed curves with prescribed curvature

本文利用变分方法并结合Struwe的单调性技巧，通过引入参数获得有界Palais-Smale序列，从而证明某些曲率函数类下具有指定曲率的平面闭曲线的存在性。

Finiteness of the total first curvature of a non-closed curve in $\mathbb{E}^{n}$

We consider a regular smooth curve in $\mathbb{E}^n$ such that its coordinates' components are the fundamental solutions of the differential equation $ y^{(n)} (x) - y(x) = 0 ,$ $x \in \mathbb{R} $ of order $n$. We show that the total first curvature of this curve is infinite for odd $n$ and is finite for even $n$.

解析曲面上的不朽曲线缩短流

Immortal Curve Shortening Flow on Analytic Surfaces

本文证明在闭可定向实解析黎曼曲面上，任何光滑浸入的闭不朽曲线缩短流均收敛至唯一的非常数闭测地线。

Affine subspaces of curvature functions from closed planar curves

Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+λf$ to be the curvature in the arc-length of a closed planar curve for all real $λ$. Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied. Finally, the characterization obtained is used to show that a sufficient analogue of the 4-vertex theorem cannot be developed.

Optimal homeomorphisms between closed curves

The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance.

Length-constrained curve diffusion

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow and that the only closed rotators are circles.

The geometry of canal surfaces and the length of curves in de Sitter space

We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a lower bound for the total conformal torsion of closed space curves.

On the unfolding of simple closed curves

I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the result of Connelly, Demaine, and Rote that such a motion exists for polygons. I also formulate a generalization of their program, thereby making steps toward a fully continuous proof of the result. To facilitate this, I generalize two of the primary tools used in their program: th…

Closed strong spacelike curves, Fenchel theorem and Plateau problem in the 3-dimensional Minkowski space

We generalize the Fenchel theorem for strong spacelike closed curves of index $1$ in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to $2π$. Here strong spacelike means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve $γ$ under consideration. The assumption of index 1 is equivalent to saying that $γ$ winds around some timelike axis with winding number 1. We prove this reversed Fenchel-type inequality by constructing a ruled spacelike surface with the given curve as boundary and applying the Gaus…

带奇点的闭曲线的全绝对曲率

The total absolute curvature of closed curves with singularities

研究将 Fenchel 定理推广至 $\mathbb{R}^n$ 中带奇点的闭 frontal，给出全绝对曲率下界及等号情形，并刻画尖点数量与简单性的关系。

On closed finite gap curves in spaceforms II

We prove that the set of closed finite gap curves in hyperbolic 3-space $\mathbb{H}^3$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{H}^3$. We also show that the set of closed finite gap curves in any 2-dimensional space form $\mathbb{E}^2$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{E}^2$.

On Closed Finite Gap Curves in Spaceforms I

We show that the spaces of closed finite gap curves in ${\mathbb R}^3$ and ${\mathbb S}^3$ are dense with respect to the Sobolev $W^{2,2}$-norm in the spaces of closed curves in ${\mathbb R}^3$ respectively ${\mathbb S}^3$.

Closed planar curves without inflections

We define a computable topological invariant $μ(γ)$ for generic closed planar regular curves $γ$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $μ(γ)$ on a given $γ$.

A simple closed curve in $\mathbb{R}^3$ whose convex hull equals the half-sum of the curve with itself

If $Γ$ is the range of a Jordan curve that bounds a convex set in $\mathbb{R}^2,$ then $\frac{1}{2}(Γ+Γ)=\mathsf{co}(Γ),$ where $+$ is the Minkowski sum and $\mathsf{co}$ is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in $\mathbb{R}^3$ with range $Γ$ such that $\frac{1}{2}(Γ+Γ)=[0,1]^3=\mathsf{co}(Γ).$ Also we show that such simple closed curve cannot be rectifiable.

通过极小极大方法存在常曲率闭合嵌入曲线

Existence of closed embedded curves of constant curvature via min-max

本文研究闭合定向黎曼曲面上预定测地曲率泛函的 Almgren-Pitts 极小极大理论，给出产生常曲率闭合嵌入曲线的条件，并证明在 1/8-pinched 高斯曲率的任意 $S^2$ 度量上存在任意预定常曲率的闭合嵌入曲线。

On Area-preserving and Length-preserving Nonlocal Flow of Convex Closed Plane Curves

For any $α>0,$ we study $k^α$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty}% $-norm.$\ $Other relevant $k^α$-type nonlocal flow is also discussed when $α\geq1.\ $

The spaces of non-contractible closed curves in compact space forms

We calculate the rational equivariant cohomology of the spaces of non-contractible loops in compact space forms and show how to apply these calculations for proving the existence of closed geodesics.

A gradient flow for the p-elastic energy defined on closed planar curves

We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$, that is in the case of the classic elastic energy, long-time existence is retrieved.

Shortest closed curve to inspect a sphere

We show that in Euclidean 3-space any closed curve which lies outside the unit sphere and contains the sphere within its convex hull has length at least $4π$. Equality holds only when the curve is composed of $4$ semicircles of length $π$, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996.

Tri-hamiltonian Toda lattice and a canonical bracket for closed discrete curves

Flows on (or variations of) discrete curves in $\R^2$ give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierachy. In the paper it is shown that the canonical symplectic structure on $\R^{2N}$, which can be interpreted as the phase space of closed discrete curves in $\R^2$ with length $N,$ induces Poisson commutation relations on the above mentioned subalgebra which yield the tri-hamiltonian poisson structure of the Toda lattice hierachy.

On the curve diffusion flow of closed plane curves

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 which develop at least one singularity in finite time and initially embedded curves which 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 sufficien…

Tangent lines, inflections, and vertices of closed curves

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, where P' is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plan…

$J$-holomorphic curves from closed $J$-anti-invariant forms

We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher dimensional analogue is established. We also show the dimension of closed $J$-anti-invariant $2$-forms on an almost complex $4$-manifold is a birational invariant, in the sense that it is invariant under degree one pseudoholomorphic maps.

Deformation Minimal Bending of Compact Manifolds: Case of Simple Closed Curves

The problem of minimal distortion bending of smooth compact embedded connected Riemannian $n$-manifolds $M$ and $N$ without boundary is made precise by defining a deformation energy functional $Φ$ on the set of diffeomorphisms $\diff(M,N)$. We derive the Euler-Lagrange equation for $Φ$ and determine smooth minimizers of $Φ$ in case $M$ and $N$ are simple closed curves.

Geometric filling curves on surfaces

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

Uniqueness of Area Minimizing Surfaces for Extreme Curves

Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simple closed curves in the boundary of M which bound unique absolutely area minimizing surfaces in M is dense in the space of simple closed curves in the boundary of M which are nullhomologous in M.

Vertices of closed curves in Riemannian surfaces

We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.

Closed trajectories of a particle model on null curves in anti-de Sitter 3-space

We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.

On the existence of closed $C^{1,1}$ curves of constant curvature

We show that on any Riemannian surface for each $0<c<\infty$ there exists an immersed $C^{1,1}$ curve that is smooth and with curvature equal to $\pm c$ away from a point. We give examples showing that, in general, the regularity of the curve obtained by our procedure cannot be improved.

闭复曲线上手性 de Rham 复形的整体截面

The Global Sections of Chiral de Rham Complexes on Closed Complex Curves

计算了亏格 $g \ge 2$ 的闭复曲线上手性 de Rham 复形的整体截面空间。

Topology and Closed Timelike Curves II: Causal structure

Because no closed timelike curve (CTC) on a Lorentzian manifold can be deformed to a point, any such manifold containing a CTC must have a topological feature, to be called a timelike wormhole, that prevents the CTC from being deformed to a point. If all wormholes have horizons, which typically seems to be the case in space-times without exotic matter, then each CTC must transit some timelike wormhole's horizon. Therefore, a Lorentzian manifold containing a CTC may nevertheless be causally well behaving once its horizon's are deleted. For instance, there may be a Cauchy-like surface through wh…