Jan L. Cieslinski
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Two special cases, namely dicrete pseudospherical surfaces and smooth pseudosperical surfaces are consistent with this description. In particular, we define the Gaussian curvature in the discrete case.
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On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature (pseudospherical) surfaces in Euclidean three space. We also compute two examples by this formula in detail.
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gaussian curvature which turns out to be valid for asymptotic nets on any time scale. We show that asymptotic Chebyshev nets on an arbitrary time scale have constant negative Gaussian curvature. We present also the quaternion-valued spectral problem (the Lax pair) and the Darboux-Back…
We apply the loop group method developed by Zakharov-Shabat, Terng-Uhlenbeck and Toda to the study of symmetries of pseudospherical surfaces in R^3. In this paper (part I) we consider the general theory, while in a second paper (part II) we will study special cases.
We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solutio…
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equations occupy a particularly special place amongst pss equations, since a series of recent papers [2, 7, 11, 12, 13], on several classes of pss equations, seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples are given, which include the sh…
The paper presents a generalized Weierstrass representation for pseudospherical surfaces in terms of 3x3 matrices, using moving frames and loop group decompositions. The construction of all such surfaces, starting from a given representation, constitutes the subject of a separate, ulterior report. The appendix of the present work gives an elementary proof of the Birkhoff decomposition theorem for "loop-groups" of spectral parameter based on the real line, instead of the unit circle.
A look on equations describing pseudospherical surfaces
本文回顾了伪球面曲面描述方程的发展，从Sasaki到当前柯西问题及几何影响的研究。
Second order partial differential equations which describe spherical surfaces (ss) or pseudospherical surfaces (pss) are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K = 1$ or $K = -1$, respectively, and they can be seen as the compatibility condition of an associated su(2)-valued or sl(2, R)-valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we give a complete and explicit classiffcation of equations of the form $z_{tt} = A(z, z_x , z_t) z_{xx} + B(z, z_x , z_t ) z_{xt} + C(z, z_x …
The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa-Holm and Degasperis-Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in ${\bf E}^{3}$ of the ps…
Third order equations, which describe spherical surfaces (ss) or pseudospherical surfaces (pss), of the form \[ ν\,z_{t}-λ\,z_{xxt}=A(z,z_{x},z_{xx})\,z_{xxx}+B(z,z_{x},z_{xx}) \] with $ν$, $λ$ $\in$ $\mathbb{R}$, $ν^2+λ^2\neq 0$, are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K=1$ or $K=-1$, respectively. Alternatively they can be seen as the compatibility condition of an associated $\mathfrak{su}(2)$-valued or $\mathfrak{sl}(2,\mathbb{R})$-valued linear problem, also referred to as a zero curvature representation. Under certain …
Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems
该研究利用Kato半群方法证明描述伪球面的Novikov方程周期初值柯西问题解的适定性，进而得到周期伪球面的存在唯一性及相关几何量的正则性结果。
Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations
本文研究由一类三阶微分方程定义的伪球面，证明其在三维欧氏空间中局部等距浸入的第二基本形式三元组仅依赖于解的零阶射流，具有通用性。
Constant curvature rotational nets and periodic Bäcklund transforms
本文研究了常负高斯曲率旋转网的Bäcklund变换，构造了具有显式参数化的非旋转离散伪球面新例子，并给出了保持周期性的条件。
Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surface…
The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vortex filament flow (also known as the Localized Induction Equation). In doing so, we obtain analytic constant-torsion representatives for a large number of knot types.
Intrinsic geometry and wave-breaking phenomena in solutions of the Camassa-Holm equation
该研究聚焦Camassa-Holm方程柯西问题对应的伪球面，分析全局解对曲面的影响与两类度量奇点，证明度量爆破等价于解在有限时间内破缺，为理解该方程的破波现象提供几何视角。
It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with ano…
A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter space-time are obtained in order to illustrate this correspondence.
We determine all tight Lagrangian surfaces in $S^2 \times S^2$. In particular, globally tight Lagrangian surfaces in $S^2 \times S^2$ are nothing but real forms.
We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.
Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and to characterize them using a geometric invariant called lightlike curvatures. As a result, we reveal the relationship between timelike Thomsen surfaces, and timelike minimal surfaces with planar curvature lines. As an application, we give a deformation of null curves preserving …
We classify non-minimal biconservative surfaces with parallel mean curvature vector field in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$. When these surfaces do not lie in $\mathbb{S}^n$ or $\mathbb{H}^n$ and they are not vertical cylinders, we find their explicit (local) equation. We also prove a result on the compactness of biconservative surfaces with constant mean curvature in Hadamard manifolds.
The aim of this paper is to extend classic results of the theory of CMC surfaces in the product spaces to the class of immersed surfaces in $\mathbb{M}^2(κ)\times\mathbb{R}$ whose mean curvature is given as a $C^1$ function depending on their angle function. We cover topics such as the existence of a priori curvature estimates for graphs, height estimates for horizontal and vertical compact graphs and a structure-type result, which classifies all the simply connected, properly embedded surfaces with finite topology.
We apply the complex analysis over the double numbers $D$ to study the minimal time-like surfaces in $R^4_2$. A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit special geometric (canonical) parameters. Then the geometry of the minimal time-like surfaces of general type is determined by the Gauss curvature $K$ and the curvature of the normal connection $\varkappa$, satisfying the system of natural equations for these surfaces. We prove the fol…
In this paper we classify constant angle surfaces in $\H^2\times\R$, where $\H^2$ is the hyperbolic plane.
In this article we study surfaces in $\mathbb{S}^3(1) \times \mathbb{R}$ for which the $\mathbb{R}$-direction makes a constant angle with the normal plane. We give a complete classification for such surfaces with parallel mean curvature vector.
We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first fundamental form are expressed by the invariants of the surface. On any time-like surface we introduce a special complex function over the algebra of the double numbers and apply the analysis over the algebra of the double numbers as a convenient tool to study these surfaces. Then the canonical coordinates on minimal time-like surfaces are characterized by a natural c…
In this article we consider surfaces in the product space $\h^2\times \r$ of the hyperbolic plane $\h^2$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete minimal graphs; the classification of umbilical surfaces.
In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvature and investigate their properties.
We deal with the minimal Lagrangian surfaces of the Einstein-Kähler surface $S^2 \times S^2$, studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in $S^3 \subset R^4$. We also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour.
We classify (spacelike or timelike) surfaces of revolution with zero $f$-mean curvature in $\Bbb G^2\times\Bbb R_1,$ the Lorentz-Minkowski 3-space $\Bbb R^3_1$ endowed with the Gaussian-Euclidean density $e^{-f(x,y,z)}=\frac 1{2π}e^{-\frac{x^2+y^2}2}.$ It is proved that an $f$-maximal surface of revolution is either a horizontal plane or a spacelike $f$-Catenoid. For the timelike case, a timelike $f$-minimal surface is either a vertical plane containing $z$-axis, the cylinder $x^2+y^2=1,$ or a timelike $f$-Catenoid. Spacelike and timelike $f$-Catenoids are new examples of $f$-minimal surfaces …
In this paper we introduce the notion of timelike surface with harmonic inverse mean curvature in 3-dimensional Lorentzian space forms, and study their fundamental properties.
We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface we introduce locally natural principal parameters and prove that such a surface is determined uniquely (up to motion) by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution of the Lund-Regge reduction problem for time-like W-surfaces with real principal curvatures in Minkowski space. We apply this theory to linear fractional time-like W-surfaces and obtain the natur…
In this article, we study complete surfaces $Σ$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature $K_e$. Let $K_i$ denote the intrinsic curvature of $Σ$. Assume that the equation $aK_i+bK_e=c$ holds for some real constants $a\neq0$, $b>0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational.
Long-time behavior of Ricci flow on some complex surfaces
本文通过双Lipschitz模型研究了某些4维流形（一般型极小曲面）上Ricci流的长时间行为，展示了膨胀与静态行为的结合。
In this paper we study rotational surfaces in the space $\mathbb{H}^2\times\mathbb{R}$ whose mean curvature is given as a prescribed function of their angle function. These surfaces generalize, among others, the ones of constant mean curvature and the translating solitons of the mean curvature flow. Using a phase plane analysis we construct entire rotational graphs, catenoid-type surfaces, and exhibit a classification result when the prescribed function is linear.
We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$ via conjugate constructions. The resulting surfaces can be seen as the analogy to the Schwarz P-surface in these homogeneous 3-manifolds.
Geodesic boundary of constant mean curvature surfaces in $\mathbb{H}^2\times \mathbb{R}$
将极小曲面测地边界结果推广至常平均曲率 $0\le H\le 1/2$ 的曲面，研究其在 $\mathbb{H}^2\times \mathbb{R}$ 中的边界性质。
The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(κ)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In this paper we give necessary and sufficient conditions for the existence of prescribed mean curvature spheres, and we describe complete surfaces of revolution proving that they behave as the Delaunay surfaces of CMC type.
We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When $n=2$, our results are similar to those obtained in \cite{bds} for surfaces with constant mean curvature in space forms.
In this paper almost complex surfaces of the nearly Kähler $S^3\times S^3$ are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kähler $S^3\times S^3$. We also find a correspondence between almost complex surfaces in the nearly Kähler $S^3\times S^3$ and solutions of the general $H$-system equation introduced by Wente, thus obtaining a geometric interpretation of solutions of the general $H$-system equation. From this we deduce a correspondence between constant m…
The goal of this article is to study minimal surfaces in $\mathbb{M}^2 \times \mathbb{R}$ having finite total curvature, where $\mathbb{M}^2$ is a Hadamard manifold. The main result gives a formula to compute the total curvature in terms of topological, geometrical and conformal data of the minimal surface. In particular, we prove the total curvature is an integral multiple of $2π$.
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.
Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces
该研究证明直纹面上有限时间坍塌Kähler-Ricci流产生以$\text{P}^1\times \text{C}$为模型的I型奇点，并由此得到直纹面纤维的最优坍塌速率。
Given $H\in [0,\infty),$ some sufficient conditions for existence of CMC $H$ graphs with boundary in two parallel planes of $\mathbb{H}^2\times\mathbb{R}$ are presented. Height estimates for outwards-oriented CMC surfaces (horo)cyllindrically bounded are also exhibited.
Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I
本文证明紧致Kähler曲面上有限时间坍缩的Kähler-Ricci流奇点均为Type I型，结合已有结果，所有有限时间奇点均为Type I型。
Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants
本文证明在Sobolev常数一致有界时，紧致Kähler曲面上的Calabi流可越过任意有限时间延拓，并在初始Calabi能量低于显式阈值时得到长时间存在性。
There are examples of complete spacelike surfaces in the Lorentzian product $\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K\leq -1$. In this paper, we show that there exists no complete spacelike surface in $\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K>-1$.
The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [Izumiya, S. and Romero Fuster, M. C. Selecta Mathematica (NS), 13 23--55 (2007)] which is a natural Lorentzian analogue of the classical Euclidean differential geometry of hypersurfaces. In this paper we investigate a special class of surfaces (i.e., marginally trapped surfaces) in Minkowski space-time from the view point of the lightlike geometry.
We consider surfaces with parallel mean curvature vector (pmc surfaces) in $\mathbb{C}P^n\times\mathbb{R}$ and $\mathbb{C}H^n\times\mathbb{R}$, and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic differential on such surfaces. This is then used in order to show that the anti-invariant pmc $2$-spheres of a $5$-dimensional non-flat cosymplectic space form of product type are actually the embedded rotational spheres $S_H^2\subset\bar M^2\times\mathbb{R}$ of Hsiang and Pedrosa, where $\bar M^2$ is a complete simply-connected surface with constant curvature. When …
We prove that every complete connected immersed surface with positive extrinsic curvature $K$ in $H^2\times R$ must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature ($K-$surfaces). We establish that the only complete $K-$surfaces in $S^2\times R$ and $H^2\times R$ are rotational spheres. Here are the key steps to achieve this. First height estimates for compact $K-$surfaces in a general ambient space $M^2\times R$ with boundary in a slice are obta…
We consider the asymptotic behavior of properly embedded minimal surfaces in the product of the hyperbolic plane with the line, taking into account the fact that there is more than one natural compactification of this space. This provides a better setting in which to consider the general problem of determining which curves at infinity are the asymptotic boundary of such minimal surfaces. We also construct some new examples of such surfaces and describe the boundary regularity.
We construct non-zero constant mean curvature H surfaces in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height and are invariant under a discrete group of horizontal translations. In $\mathbb{S}^2\times\mathbb{R}$ (for any $H > 0$) or $\mathbb{H}^2\times\mathbb{R}$ (for $H > 1/2$), a 1-parameter family of unduloid-type surfaces is obtained, some of which are shown to be compact in $\mathbb{S}^2\times\mathbb{R}$. Finally, in the case of $H = 1/2$ in $\…
In this article, we consider compact surfaces $Σ$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $Γ=\partialΣ\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the warped product $ \mathbb{M}\times_f\mathbb{R} $, here $ \mathbb{M} $ denotes a Hadamard surface. We obtain height estimate for a such surface $Σ$ having positive constant mean curvature involving the area of a part of $Σ$ above of $ \mathbb{M} _0$ and the volume it bounds. Also we give general conditions for the existence of rotationally-invariant topological spheres havin…
Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. First, using a characterization of $δ$-parabolicity we prove that under additional conditions on $\mathbb{M}$, an embedded minimal surface with bounded gaussian curvature is proper. The second theorem states that under some conditions on $\mathbb{M}$, if $Σ$ is a properly immersed minimal surface with finite topology and one end in $\mathbb{M}\times\mathbb{R}$, which is transverse to a s…
A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquely up to a motion in space-time by the Gauss curvature and the normal curvature, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal space-like surfaces. Using canonical Weierstrass representations of minimal sp…
We study almost complex surfaces in the nearly Kähler $S^3\times S^3$. We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente. We find a global holomorphic differential on every almost complex surface, and show that when this differential vanishes, then the corresponding solution of the H-surface equation gives a constant mean curvature surface in $\mathbb{R}^3$. We use this, together with a theorem of Hopf, to classify all almost complex 2-spheres. In fact there is essentially only one, and it is totally geodes…
Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2π$. The first examples appearing in this context are vertical geodesic planes and Scherk minimal graphs over ideal polygonal domains. Other non simply-connected examples have been constructed recently. In the present paper, we show that the only complete minimal surfaces in $\H^2\times\R$ of total curvature $-2π$ are Scherk minimal graphs over ideal quadrilaterals. …
We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere holomorphic harmonic map from an open simply connected Riemann surface into the Poincare disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on the surface can be realized as the Abresch-Rosenberg holomorphic differential of some, and generic…
We define the notions of $S_t^1\times S_s^1$-valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of standard techniques of singularity theory for the Lorentzian lightcone height function.
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