Sigurd B. Angenent
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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。
We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We use this characterization to provide a qualitative study of the solitons. We show that for each fixed vector there is a 2-parameter family of soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the entire real line, it is embedded and its geodesic curvature converges to a con…
In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let $\mathbf{M}$ be a compact locally symmetric space. If the curve shortening flow exists for infinite time, and $$ \lim_{t\to\infty}L(γ_{t})>0, $$ then for every $n>0$, $$ \lim_{t\to \infty}\sup(|\frac{D^{n}T}{\partial s^{n}}|)=0. $$ In particular, the limiting curve exists and is a …
Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces
该研究刻画了三维欧氏空间中旋转曲面上曲线缩短流的旋转孤子，并证明这类开曲线的两端渐近于平行测地线，完善了相关几何流的孤子理论。
We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial curve is a graph over $S^1$, then its curve shortening flow exists for all times and finally converges to a geodesic closed curve.
From curve shortening to flat link stability and Birkhoff sections of geodesic flows
本文利用曲线缩短流证明了闭黎曼曲面测地流的三个新结果：平坦链环在 $C^0$-小扰动下的稳定性、闭测地线的强制存在性定理，以及测地流 Birkhoff 截面的存在性。
We formulate a uniqueness conjecture for curve shortening flow of proper curves on certain symmetric surfaces and give an example of a non-flat metric on the plane with respect to which curve shortening flow is not unique. That is, with respect to a suitably chosen metric, we construct a non-static solution to curve shortening flow starting from a properly embedded geodesic.
Non-uniqueness of geodesic limits and a question of Grayson and Gage
针对Grayson与Gage提出的闭曲面上不朽曲线缩短流的极限测地线是否唯一的问题，在球面$\rm S^2$上构造反例，证明该极限测地线不唯一，否定回答了该问题。
We define Type I singularities for the mean curvature flow associated to a density $ψ$ ($ψ$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $ψ$-curvature due to the density. We describe a family of curves whose evolution under $ψ$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.
We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of constant curvature. Specifically, we show that such curves do not develop singularities, and evolve to geodesics.
In this paper, we study a family of curves on $S^2$ that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective plane. In addition, as a consequence of our main result, we show that any two smooth embedded curves on $RP^2$ which intersect transversally at exactly one point converge to two different geodesics under the flow.
Conditions for the existence of closed geodesics is a classic, much-studied subject in Riemannian geometry, with many beautiful results and powerful techniques. However, many of the techniques that work so well in that context are far less effective in Lorentzian geometry. In revisiting this problem here, we introduce the notion of timelike geodesic homotopy, a restriction to geodesics of the more standard timelike homotopy (also known as $t$-homotopy) of timelike loops on Lorentzian manifolds. This tool is combined with a local shortening/stretching of length argument to provide a number of n…
In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.
A novel finite element approximation of anisotropic curve shortening flow
本文将 DeTurck 技巧推广至空间依赖的各向异性曲线缩短流，提出新弱形式，证明半离散有限元最优 $H^1$ 误差界与全离散格式无条件稳定性，并通过数值实验验证。
We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics. In order to prove the first theorem, we employ the generalization of Grayson's curve shortening flow developed by Angenent-Oaks.
Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary
本研究通过分析带严格凸边界闭曲面上的自由边界曲线缩短流的非坍缩性与收敛性，证明了带严格凸边界的黎曼2-圆盘上存在两条自由边界嵌入测地线及特定Morse指数的简单闭测地线，解决了Grayson定理的自由边界对应问题。
In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. In the present paper this affine method is refined and adapted to the pseudo-Riemannian case. Using this method and elementary topology it is proved that any homogeneous Lorentzian manifold of even dimension admits a light-like homogeneous geodesic. The method is illustrated in d…
Random 3-Manifolds Have No Totally Geodesic Submanifolds
本文证明在紧致三维流形上，不具有全测地子流形的黎曼度量集合包含一个在C^q拓扑（q≥3）中开且稠密的子集，从而加强了Lytchak和Petrunin关于三维情形通用集仅为稠密G_δ集的结果。
Surfaces of section for geodesic flows of closed surfaces
本文基于曲线缩短流证明了闭可定向黎曼曲面测地流的截面存在性，并研究了简单闭测地线的构型。
In this paper we prove that a properly embedded constant mean curvature surface in $\mathbb{H}^2\times\mathbb{R}$ which has finite topology and stays at a finite distance from a vertical geodesic line is invariant by rotation around a vertical geodesic line.
We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically nite manifolds.
The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius only depending on the topology of the surface (and the size of the first embedded disk), which is disjoint from any simple closed geodesic.
On the Cut Locus of Submanifolds of a Finsler Manifold
本文研究前向完备Finsler流形中闭子流形的割迹，通过引入N-测地环概念，将Klingenberg闭测地线引理推广到可逆Finsler情形下的N-测地环。
The goal of this survey is to give a list of resent results about topology of manifolds admitting different metrics with the same geodesics. We emphasize the role of the theory of integrable systems in obtaining these results.
We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu. (We do not use a Lusternik-Schnirelmann type theorem on the existence of at least three simple closed geodes…
The space ML(F) of measured geodesic laminations on a given closed hyperbolic surface F has a canonical linear structure arising in fact from different sources in 2-dimensional hyperbolic (earthquake theory) or complex projective (grafting) geometry as well as in (2+1) Lorentzian one (globally hyperbolic spacetimes of constant curvature). We investigate this linear structure, by showing in particular how heavily it depends on the geometric structure of F, whle to many other extens ML(F) depends only on the topology of F. This is already manifest when we describe in geometric terms the sum of t…
We obtain simple characterizations of the connected components of the space of closed curves on the 2-sphere whose geodesic curvatures are constrained to lie in an open interval $(κ_1,κ_2)$, in terms of $κ_1$ and $κ_2$. Many results concerning the topology of these spaces are established. In particular, we determine the homeomorphism class of some of these components.
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…
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classi…
Bifurcations of magnetic geodesic flows on surfaces of revolution
本文研究2-球面上旋转不变的磁测地流可积系统，刻画了其Liouville纤维化拓扑、奇点类型，计算了Fomenko--Zieschang不变量，并完整描述了动量映射的分岔图结构。
Extensions of the loop product and coproduct, the space of antipodal paths and resonances of closed geodesics
本文推广弦拓扑中的环积与环余积至对合确定的路径空间，并在偶数维球面上利用 Morse-Bott 方法完整计算，进而证明实射影空间闭测地线的共振定理。
In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the $C^0$ topology. We establish several instances of entropy robustness (persistence of entropy non-vanishing after small $C^0$ perturbations). A large part of this paper is dedicated to metrics on the 2-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy (thus generalizing and quantifying a result of Denvir-Mackay) and that metrics with robust positive entropy on the torus are $C^{\infty}$ generic. More…
Froyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in three and four-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of mini…
Given a manifold $M$ and a proper sub-bundle $Δ\subset TM$, we study homotopy properties of the horizontal base-point free loop space $Λ$, i.e. the space of absolutely continuous maps $γ:S^1\to M$ whose velocities are constrained to $Δ$ (for example: legendrian knots in a contact manifold). A key technical ingredient for our study is the proof that the base-point map $F:Λ\to M$ (the map associating to every loop its base-point) is a Hurewicz fibration for the $W^{1,2}$ topology on $Λ$. Using this result we show that, even if the space $Λ$ might have deep singularities (for example: constant lo…
We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。