The homotopy type of the space of degree 0 immersed plane curves

Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.

Homotopy Groups of the Space of Curves on a Surface

We explicitly calculate the fundamental group of the space $\mathcal F$ of all immersed closed curves on a surface $F$. It is shown that $π_n(\mathcal F)=0$, n>1 for $F\neq S^2, RP^2$. It is also proved that $π_2(\mathcal F)=\Z$, and $π_n(\mathcal F)=π_n(S^2)\oplusπ_{n+1}(S^2)$, n>2, for $F$ equal to $S^2$ or $RP^2$.

Existence of minimizing Willmore Klein bottles in Euclidean four-space

Let $K$ be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles in Euclidean $n$-space is attained by a smooth embedded Klein bottle, where $n\geq 4$. There are three distinct regular homotopy classes of immersed Klein bottles in the Euclidean four-space each one containing an embedding. One is characterized by the property that it contains the minimizer just mentioned. For the other two regular homotopy classes we show that the Willmore energy is bounded from below by $8π$. We give a classification of the minimizers of these two classes. In particul…

On the Lagrangian angle and the Kähler angle of immersed surfaces in the complex plane $\bbc^2$

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kähler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kähler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kähler angle which imply rigidity theorems of self-shr…

Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding-Minicozzi's notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.

Properly embedded and immersed minimal surfaces in the Heisenberg group

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We give an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs…

On minimal surfaces immersed in three dimensional Kropina Minkowski space

In this paper we consider a three dimensional Kropina space and obtain the partial differential equation that characterizes a minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface.

正曲率流形的第一和第二同伦群

On the first and second homotopy groups of manifolds with positive curvature

本文研究正曲率闭流形的同伦群约束，证明奇数维情形不存在同时在第一、第二同伦群上非平凡的实射影平面映射，且四维倍数维流形的基本群在第二同伦群上作用平凡。

Cayley cones ruled by 2-planes: desingularization and implications of the twistor fibration

Cayley cones in the octonions $\mathbb{O}$ that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration $G(2,8) -> S^6$ is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\gro$. Equivalently, this produces Cayley cones whose links are $S^1$-bundles over genus-$g$ Riemann surfaces. When the degree of an immersed pseudoholomorphic curve is large enough, the corresponding 2-ruled Cayley cone is the asymptotic cone of a non-conical 2-ruled Cayley 4-fold.

Counting the minimal number of inflections of a plane curve

Given a plane curve $γ: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(γ)$ of inflections which curves $\mbox{diff}(γ)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of $\mathbb R^2$. We show that if $γ$ is an immersed curve with $D(γ)$ double points and no other singularities, then $I(γ)\leq 2D(γ)$. In fact, we prove the latter result for the so-called plane doodles which are finite collections of closed immersed plane curves whose only singularities are double points.

Affine metrics of locally strictly convex surfaces in affine 4-space

We introduce a new family of affine metrics on a locally strictly convex surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if $M$ is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal to the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes.

平面渐近的翻译器

Translators Asymptotic to Planes

本文证明了垂直平面是唯一满足特定条件的完整翻译器。

Homotopical and topological rigidity of hypersurfaces of spherical space forms

The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the principal curvatures of such a hypersurface $ f \colon N^n \to M^{n+1} $ $( n\ge 2 )$, it asserts that the universal cover of $ N $ must be diffeomorphic to the $ n $-sphere $ {S}^n $, and provides an upper bound for the order of the fundamental group of $ N $ in terms of that of $ M $. In particular, if $ M = {S}^{n+1} $, then $ N $ is diffeomorphic to $ {S}^n …

Anisotropic curvature flow of immersed networks

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2$ norm of the anisotropic curvature blows up.

Spacelike capillary surfaces in the Lorentz-Minkowski space

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and boundary umbilic points and applying the Poincaré-Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than $4$ vertices in a domain of $\Bbb L^3$ bounded by (spacelike or timelike) totally umbilic surfaces is part of a (spacelike) pla…

Half-space theorems for minimal surfaces in Nil_3 and Sol_3

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol_3 that lies on one side of a special plane, then S is another special plane.

Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow

Let $X_0, \widetilde{X}$ be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve $X_0$ into $\widetilde{X}$. It is proved that this flow exits globally, preserves both the local convexity and the elastic energy of the evolving curve. If the two curves have same elastic energy then the curvature flow deforms the evolving curve into the target curve $\widetilde{X}$ as time tends to infinity.

复射影平面的SYZ镜像的多值Morse同伦

Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane

提出多值Morse同伦范畴的定义，并证明其全子范畴与复射影平面上的全纯向量丛范畴的A_∞等价，应用于全纯切丛的整体截面的镜像描述。

On the Global Structure of Hopf Hypersurfaces in Complex Space Form

It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed C^(2n-1) regular Hopf hypersurface is a geodesic hypersphere

Hitchin流形中测地平面的分形闭包

Fractal closures of geodesic planes in Hitchin manifolds

该研究通过构造 Hitchin 流形中的漂浮测地平面，证明了其闭包为分形，揭示了高秩无限体积情况下非紧类型局部对称空间的拓扑刚性反例。

Minimal annuli with constant contact angle along the planar boundaries

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

三维球面由简单闭曲线构成的纤维化空间的同伦型

Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves

本研究确定了三维球面简单闭曲线光滑纤维化模空间的同伦型，证明其与Hopf纤维化子空间同伦型一致，还刻画了该模空间的Fréchet流形结构与建模空间。

平坦曲面上具有约束曲率的曲线空间的同伦型

Homotopy type of spaces of curves with constrained curvature on flat surfaces

本文确定了平坦曲面上具有约束曲率且端点方向固定的曲线空间的同胚类，证明其每个连通分量可缩或同伦等价于球面，并构造显式同伦等价。

曲线缩短流中凸投影的奇点

Singularities of Curve Shortening Flow with Convex Projections

研究证明了在R^n中具有单射凸投影到某个2平面的光滑闭合浸入曲线在曲线缩短流下会产生I型奇点并渐近趋于圆形，同时证明了在R^n中曲线缩短流下的Huisken猜想的类比形式，即任何光滑闭合浸入曲线可通过光滑扰动转化为在R^{n+2}中收缩至圆点的闭合浸入曲线。

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…

Homotopy types of Diffeomorphism groups of noncompact 2-manifolds

Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the homotopy type of the circle if M is the plane, the open annulus or the open M"obius band, and it is contractible in all other cases. When M admits a volume form w, we also discuss the topological type of the group of w-preserving diffeomorphisms of M. To obtain these results we stud…

A Lie algebra structure on variation vector fields along curves in $2$-dimensional space forms

A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation is finally discussed from this point of view.

Minimal ends in H2xR with finite total curvature and a Schoen type theorem

In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal complete end E with finite total curvature is properly immersed and that the Gaussian curvature of E is locally bounded in terms of the geodesic distance to its boundary.

$R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms. We find isometries (called $R$-transforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We also show how to utilise the isometries to compute geodesics numerically.

Non-compact arithmetic manifolds have simple homotopy type

We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham factors, there are constants a=a(S) and d=d(S) such that any non-compact arithmetic manifold, locally isometric to S, is homotopically equivalent to a simplicial complex whose vertices degrees are bounded by d, and its number of vertices is bounded by a times the Riemannian volume. It is very likely that such a result holds also for compact arithmetic manifold…