Josef Dorfmeister, Junichi Inoguchi, Magdalena Toda
We derive a correspondence between (Lorentzian) harmonic maps into the pseudosphere $S_1^2$, with appropriate regularity conditions, and certain connection 1-forms. To these harmonic maps, we associate a representation of type Weierstrass, and we apply it to construct timelike surfaces with constant mean curvature.
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Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in Minkowski 3-space. The relationship between timelike minimal surfaces and bosonic Nambu-Goto string worldsheets in spacetime is also discussed in the appendix.
A Liouville-Weierstrass correspondence for Spacelike and Timelike Minimal Surfaces in $\mathbb{L}^3$
本文研究了Liouville方程与空间状和时间状最小曲面在$\mathbb{L}^3$中的对应关系，通过复分析和para复分析提供了统一处理方法，并给出了显式例子。
Bour's minimal surface has remarkable properties in three dimensional Minkowski space. We reveal the definite and indefinite cases of the Bour's surface using Weierstrass representations, and give some differential geometric properties of the astonishing maximal and minimal surfaces.
Timelike surfaces in the three-dimensional Heisenberg group with left invariant semi-Riemannian metric are studied. In particular, non-vertical timelike minimal surfaces are characterized by the non-conformal Lorentz harmonic maps into the de Sitter two phere. On the basis of the characterization, the generalized Weierstrass type representation will be established through the loop group decompositions.
Weierstrass representations of discrete constant mean curvature surfaces in isotropic space
该研究获得三维迷向空间中离散常平均曲率曲面的Weierstrass表示，还借助该表示构造出带离散闭式参数化的相关实例。
We study manifolds with split-complex structure and apply some general results to the study of Lorentz surfaces. In particular, we apply our results to timelike minimal immersions. The conformal realization of these surfaces is obtained using a representation based on loop groups. The classical Weierstrass representation (integral formula) is recovered as a byproduct of this general setting.
The study of the relation between the Weierstrass inducing formulae for constant mean curvature surfaces and the completely integrable euclidean nonlinear sigma-model suggests a connection among integrable sigma -models in a background and other type of surfaces. We show how a generalization of the Weierstrass representation can be achieved and we establish a connection with the Weingarten surfaces. We suggest also a possible generalization for two-dimensional surfaces immersed in a flat space R^8 with Euclidean metric.
Let $(M,\mathsf{d},\mathfrak{m},\ll,\leq,τ)$ be a causally closed, $\mathscr{K}$-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition $\smash{\mathrm{wTCD}_p^e(K,N)}$ in the sense of Cavalletti and Mondino. We prove the existence of geodesics of probability measures on $M$ which satisfy the entropic semiconvexity inequality defining $\smash{\mathrm{wTCD}_p^e(K,N)}$ and whose densities with respect to $\mathfrak{m}$ are additionally uniformly $L^\infty$ in time. This holds apart from any nonbranching assumption. We also discu…
For all $k\in]0,1[$, we construct a canonical bijection between the space of ramified coverings of the sphere and the space of complete immersed surfaces in $3$-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to $k$. We show, furthermore, that this bijection restricts to a homeomorphism over each stratum of the space of ramified coverings of the sphere.
Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of order 3; they are not timelike Jordan Osserman. They are k-spacelike higher order Jordan Osserman for $2\le k\le s$; they are k-timelike higher order Jordan Osserman for $s+2\le k\le 2s$, and they …
Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
本文在洛伦兹长度空间中引入双曲角概念，用于刻画类时曲率界并通过角度单调性条件实现，同时改进了测地线非分支结果。
The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature
该研究针对含完整类时线、非负类时曲率的全局双曲洛伦兹长度空间，通过构造类时渐近线、引入类时线平行性概念，证明了综合分裂定理，并将局部分裂结果推广至全局。
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.
In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface which admits some kind of singular points.
We exhibit a family of generalized plane wave manifolds of signature (2,2). The geodesics in these manifolds extend for infinite time (i.e. they are complete), they are spacelike and timelike Jordan Osserman, and they are spacelike and timelike Jordan Ivanov-Petrova. Some are irreducible symmetric spaces. Some are homogeneous spaces but not symmetric. Some are 1-curvature homogeneous but not homogeneous. All are 0-modeled on the same irreducible symmetric space. We determine the Killing vector fields for these manifolds.
We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons w…
Causal convergence conditions through variable timelike Ricci curvature bounds
该研究提出非光滑整体双曲正则长度度量时空概念，引入带可变Ricci曲率界的熵类时曲率维数条件，统一广义相对论能量条件，推广相关几何估计与奇点定理，给出稳定性结果与类时几何不等式。
A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representations - canonical Weierstrass representations via two holomorphic functions. We find the expressions of the Gauss curvature and the normal curvature of the surface with respect to this pair of holomorphic functions. We find the relation between two pairs of holomorphic functions ge…
In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space. Firstly, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the Mannheim offsets of a timelike ruled surface may be timelike or spacelike. Furthermore, we characterize the developable of Mannheim offset of a timelike ruled surface by the derivative of the conical curvature of the directing cone.
On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this system determine a unique (up to a motion) timelike surface with zero mean curvature so that the given parameters are canonical. We find all timelike surfaces with zero mean curvature in the class of rotational surfaces of Moore type. These examples give rise to a one-parameter f…
It is shown that timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in PSL(2,R) via Bryant type representation formulae. These formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson correspondence, between timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space and timelike minimal surfaces in Minkowski 3-space. The hyperbolic Gauss map of timelike surfaces in anti-de Sitter 3-space, which is a close analogue of the classical…
The equivalence between timelike Ricci curvature and the timelike Brunn Minkowski inequality on synthetic Lorentzian spaces
本文研究了合成洛伦兹空间中时序里奇曲率与时序Brunn-Minkowski不等式的关系，证明了在时序q-非分支设定下，q-时序曲率维度条件与时序Brunn-Minkowski条件等价，扩展了在非光滑设定下的研究成果。
Total curvature and length estimates for timelike curves in Lorentzian length spaces
本文在具有上曲率界的洛伦兹长度空间中引入类时曲线的合成总曲率概念，证明其与光滑情形一致，并给出类时曲线长度的尖锐下界。
In this paper, we study the timelike tubular Weingarten surfaces in 3-dimensional Minkowski space $IR_1^3 $.We have obtained some conditions for being $({K_{II},H})$, $({K_{II},K})$, timelike tubular Weingarten surfaces where are the second Gaussian curvature the Gaussian curvature and the mean curvature, respectively.
We construct new integrable systems to present Weierstrass type representations for spacelike surfaces whose mean curvature vector $\mathbf{H}$ satisfies the null condition $\langle \mathbf{H}, \mathbf{H} \rangle=0$ in the four dimensional Lorentz-Minkowski space ${\mathbb{L}}^{4}$. Our new Weierstrass presentations extend simultaneously classical Weierstrass representations (of the first kind and the second kind) for maximal surfaces in ${\mathbb{L}}^{3}$ and minimal surfaces in ${\mathbb{R}}^{3}$. We solve a linear partial differental equation to construct explicit examples of marginally tra…
Weierstrass-type representations have been used extensively in surface theory to create surfaces with special curvature properties. In this paper we give a unified description of these representations in terms of classical transformation theory of $Ω$-surfaces.
The equivalence of smooth and synthetic notions of timelike sectional curvature bounds
本文证明了在强因果时空中的时间平面曲率界与合成时间界之间的等价性，并建立了时间分离和带符号距离函数的Hessian比较作为工具。
Convergence of Lorentzian spaces and curvature bounds for generalized cones
本文引入了洛伦兹前长度空间的收敛概念，证明了时状曲率和时状曲率-维度界在该收敛下稳定，并展示了广义洛伦兹锥在该收敛下的适应性，最终获得预紧致定理。
We give a conformal representation in terms of meromorphic data for a certain class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean curvature vector is either lightlike or zero at each point. This representation extends simultaneously the Weierstrass representation for minimal surfaces in Euclidean 3-space and for maximal surfaces in the Lorentz-Minkowski 3-space, and the Bryant representation for mean curvature one surfaces in the hyperbolic 3-space and in the de Sitter 3-space.
Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies $R_g\geq -n(n-1)$ and $\mathcal{E}\subset M$ be an asymptotically hyperbolic end, we prove that the mass functional of the end $\mathcal{E}$ is timelike future-directed or zero. Moreover, it vanishes if and only if $(M,g)$ is isometric to the hyperbolic space. We also consider the mass of an asymptotically hyperbolic manifold with compact boundary, we prove the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by a fun…
This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a blowdown argument, we show that under renormalisation this flow converges towards a homothetically expanding hyperbolic solution.
Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such parameters the Gauss curvature and the normal curvature satisfy a system of two natural partial differential equations and determine the surface up to a motion. For any minimal surface parameterized by canonical parameters we obtain Weierstrass representations - canonical Weierstra…
Timelike Meridian Surfaces of Elliptic type in the Minkowski 4-Space
本文研究 Minkowski 4-空间中位于类时轴旋转超曲面上的椭圆型类时子午面，分析了其常曲率与平行平均曲率向量场等基本类别，并为描述具有平行规范化平均曲率向量场类时曲面的自然 PDEs 背景系统提供了显式解。
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 …
In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz-Minkowski space $L^{3}$, using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples of minimal surfaces in $L^{3}$ constructed via the Enneper representation formula, that it is equivalent to the Weierstrass representation obtained by Kobayashi (for spacelike immersions) and by Konderak (for the timelike ones).
Timelike meridian surfaces of hyperbolic type in the Minkowski 4-space
研究 Minkowski 4-空间中一类特殊的类时曲面（双曲型类时子午面），证明其具有平坦法向连接，并分类了常 Gauss 曲率、极小、非零常平均曲率以及具有平行归一化平均曲率向量场的曲面。
The minimal Lorentzian surfaces in $\mathbb{R}^4_2$ whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy $K^2-\varkappa^2 >0$ are called minimal Lorentzian surfaces of general type. These surfaces admit canonical parameters and with respect to such parameters are determined uniquely up to a motion in $\mathbb{R}^4_2$ by the curvatures $K$ and $\varkappa$ satisfying a system of two natural PDEs. In the present paper we study minimal Lorentzian surfaces in $\mathbb{R}^4_2$ and find a Weierstrass representation with respect to isotherm…
A proposal of quasi-local mass for 2-surfaces of timelike mean curvature
针对现有准局域质量仅适用于视界外类空平均曲率曲面的局限，研究将Wang-Yau准局域能量/质量推广至类时平均曲率曲面（含陷获曲面），给出正定能量密度与无散流，且球对称下退化为已有定义。
In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences to the definition of timelike curvature bounds from below for Lorentzian pre-length spaces. Specifically, we establish some comparison theorems known as the local Lorentzian version of the Toponogov theorem and the Alexandrov convexity property. Finally, as an application we obtain a first variation Formula for non-negatively curved globally hyperbolic Lorentzian length spaces.
Almost-splitting of spacetimes under timelike sectional curvature bounds
该论文在类时截面曲率有下界且下界接近零的全局双曲时空中，证明了一个几乎分裂定理，表明长极大类时线段附近的因果钻石在洛伦兹Gromov--Hausdorff意义下接近乘积时空，为洛伦兹分裂定理提供了定量版本。
Minimal Timelike Surfaces in the Lorentz-Minkowski 3-space and Their Canonical Parameters
本文借助分裂复数代数中的 Weierstrass 型公式与规范参数刻画 $\mathbb R^3_1$ 极小类时曲面，建立其与全纯函数类的对应，并证明 Enneper 曲面是唯一具有三次等温多项式参数化者。
Timelike Surfaces with Parallel Normalized Mean Curvature Vector Field in the Minkowski 4-Space
本文研究 Minkowski 4-空间中具有平行归一化平均曲率向量场的类时曲面，通过引入规范参数并证明存在唯一性定理，将确定曲面的几何函数和偏微分方程个数降至最少，解决了该类曲面的 Lund-Regge 问题。
Geometric properties and offset surface construction for slant timelike-ruled surfaces
本文研究闵可夫斯基三维空间中斜类时规则曲面的几何性质，利用渐屈曲线对称性推导参数化形式，并建立中心法线与偏移曲面母线共轴的条件，通过Blaschke和Darboux标架分析曲率特征与不变量，深化对洛伦兹空间类时规则曲面微分几何的理解。
We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized su…
The scope of this survey is to give a self-contained introduction to synthetic timelike Ricci curvature bounds for (possibly non-smooth) Lorentzian spaces via optimal transport $\&$ entropy tools, including a synthetic version of Hawking's singularity theorem and a synthetic characterisation of Einstein's vacuum equations. We will also discuss some motivations arising from the smooth world and some possible directions for future research.
Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions
本研究在带测度的洛伦兹空间中引入基于Rényi熵凸性的类时曲率-维数条件与测度收缩性质，证明其相容性、稳定性、强弱等价等性质，并明确类时p-本质非分支的弱条件作用。
Stability of Synthetic Timelike Ricci Bounds under $C^0$-Limits and Applications to Impulsive Gravitational Waves
该研究证明了合成曲率-维数条件$\textsf{TCD}^e_p(K,N)$在满足一致全局双曲性时对光滑洛伦兹度量局部一致收敛稳定，并将其应用于脉冲引力波，为爱因斯坦方程低正则性解提供新几何见解。
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 …
Timelike Ollivier-Ricci curvature
本文在洛伦兹几何中引入余维数为1的粗Ricci曲率构造，通过1-Lorentz-Wasserstein距离和Moser型流恢复类时方向上的Ricci曲率及Bakry-Émery张量，呼应Raychaudhuri方程。
In this study, we investigate the existence theorems for timelike ruled surfaces in Minkowski 3-space. We obtain a general system and give the existence theorems for a timelike ruled surface according to Gaussian curvature, distribution parameter and strictional distance. Moreover, we give some special cases such as the directirx of the surface is a geodesic, an asymptotic line, a line of curvature or a general helix.
Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
本文在洛伦兹预长度空间中建立最优输运基础理论，提出类时里奇曲率下界与维数上界的合成定义并证明其弱收敛稳定性，进而将体积比较与Hawking奇点定理推广到合成框架。
A Splitting Theorem for non-positively curved Lorentzian spaces
本文提出非正曲率洛伦兹空间的分裂定理，并扩展了一阶变分公式至任意时曲率界空间。
Comparison theory for Lipschitz spacetimes
本文研究了具有局部利普希茨连续度量和下界分布性 Ricci 曲率的时空，证明其满足类时测度收缩性质，并推导出新的比较定理及体积不完整定理。
In this paper, we get the time evolution equations of the curvature and torsion of the evolving spacelike curves in the Minkowski space. Also, we give inextensible evolutions of timelike ruled surfaces that are produced by the timelike normal and spacelike binormal vector fields of spacelike curve and derive the necessary conditions for an inelastic surface evolution. Then, we compute the coefficients of the first and second fundamental forms, the Gauss and mean curvatures for timelike special ruled surfaces. As a result, we give applications of the evolution equations for the curvatures of th…
Alexandrov's Patchwork and the Bonnet--Myers Theorem for Lorentzian length spaces
本文构造了 Alexandrov 拼缀法的 Lorentz 类比，证明 Lorentz 预长度空间的局部上类时曲率界可推出全局上界，并给出类时曲率全局下界情形下的 Bonnet–Myers 型有限直径结果。
Loop Weierstrass Representation
本文提出 Loop Weierstrass 表示，统一极小曲面与常平均曲率 1 曲面的关联族、对偶曲面和 Goursat 变换，引入简单因子 dressing，并分类多种例子。
Ollivier-Ricci curvature for causal sets
利用洛伦兹最优传输引入因果集的Ollivier-Ricci曲率，通过离散化方法恢复时间型Ricci曲率，并证明局部到全局传播结果和类时Bonnet-Myers定理。
Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes
本文利用 q-Lorentz-Wasserstein 最优输运与概率测度 q-测地线，证明 Finsler 时空在加权 Ricci 曲率类时下界条件下满足 TCD_q(K,N)，并导出尖锐类时 Brunn-Minkowski 不等式。
Timelike Ricci curvature lower bounds via optimal transport for Orlicz-type Lorentzian costs
本文研究了全局超曲时空上的最优运输问题，针对Orlicz型洛伦兹成本函数u∘ℓ，提出u-分离概念并获得强对偶性，通过几何熵凸性刻画时状里奇曲率下界。
Basic Classes of Timelike General Rotational Surfaces in the Four-dimensional Minkowski Space
本文研究 Minkowski 4-空间中两类类时一般旋转曲面，刻画平坦、法联络平坦及常平均曲率等基本几何类别，并显式分类极小曲面与具有平行归一化平均曲率向量场的曲面。
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