Zhi Li, Guoxin Wei
In this paper, we obtain the classification theorem for three-dimensional complete space-like $λ$-translators $x:M^{3} \rightarrow \mathbb R^{4}_{1}$ with constant norm of the second fundamental form and constant $f_{4}$ in the Minkowski space $\mathbb R^{4}_{1}$.
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A spacelike surface in Minkowski space $\mathbb{R}_1^3$ is called a $K^α$-translator of the flow by the powers of Gauss curvature if satisfies $K^α= \langle N,\vec{v}\rangle$, $α\neq 0$, where $K$ is the Gauss curvature, $N$ is the unit normal vector field and $\vec{v}$ is a direction of $\mathbb{R}_1^3$. In this paper, we classify all rotational $K^α$-translators. This classification will depend on the causal character of the rotation axis. Although the theory of the $K^α$-flow holds for spacelike surfaces, the equation describing $K^α$-translators is still valid for timelike surfaces. So we …
Invariant $λ$-translators in Lorentz-Minkowski space
本文研究Lorentz-Minkowski空间中在1参数平移和旋转群下不变的$λ$-平移子，推导了柱面情形的显式参数化，并通过分析非线性自治系统揭示了旋转$λ$-平移子的定性性质。
The main result in this paper is a non-existence Theorem of entire $Q_{n-1}$-translators in $\mathbb{R}^{n+1}$. In addition, an example of non-entire complete $Q_{n-1}$-translator has been found and a Tangential Principle for $Q_k$-translators in $\mathbb{R}^{n+1}$.
Invariant $λ$-translators in $\mathbb{S}^2\times\mathbb{R}$
本文分类了 $\mathbb{S}^2\times\mathbb{R}$ 中由旋转单参数群和垂直平移不变的 $\lambda$-平移子，其中平均曲率满足 $H=\langle N,\partial_z\rangle+\lambda$。
Rotators-Translators to Mean Curvature Flow in $\mathbb H^2\times\mathbb R$
本文证明在双曲空间与实直线的乘积空间中，存在单参数螺旋面族，其在平均曲率流下同时绕垂直轴旋转并垂直平移。
On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$
本文对 $\mathbb H^2\times\mathbb R$ 中完备平移子的渐近边界进行分类，证明竖直边界分量为点乘区间或直线，水平边界分量为完备测地线，方法基于对称平移子屏障。
Classification of Lagrangian translators and Lagrangian self-expanders in $\mathbb{C}^{2}$
本文利用新的 Omori-Yau 型最大值原理，对 $\mathbb{C}^{2}$ 中平均曲率向量平方范数为常数的二维完备拉格朗日平移子和自扩张子进行分类，并推广到拉格朗日 $\xi$-平移子。
Translators to Higher Order Mean Curvature Flows in $\mathbb R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$
本文研究了 $\mathbb R^n\times\mathbb R$ 和 $\mathbb H^n\times\mathbb R$ 中高阶平均曲率流的平移子，证明了多种类型平移子的存在性与唯一性，并据此对常平均曲率和等参平移子进行了分类，推广了欧氏空间中的经典存在与唯一性定理。
Translators of the mean curvature flow in the special linear group $SL(2,\mathbb{R})$
本文研究特殊线性群 $SL(2,\mathbb{R})$ 中满足 $H=\langle N,X\rangle$ 的平移子，利用 Iwasawa 分解对单参数等距群不变的平移子进行分类，并对所有 Killing 向量场和不变曲面详尽讨论，部分情形给出显式参数化。
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。
Invariant $λ$-translators for the Gauss curvature flow in Euclidean space
本文针对欧氏空间中满足K=⟨N,vec{v}⟩+λ的λ-平移子，分类了所有在单参数平移群和单参数旋转群下不变的该类曲面。
On mean curvature flow translators with prescribed ends
针对Ilmanen的长期问题，构造了指标为$I$的光滑闭嵌入自收缩解$S$对应的$I$维完备平移解族，证明$\mathbb{R}^{n+1}$可分解为含此类平移解的闭集族，还发现平移解非唯一性的膨胀现象。
In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in $\mathbb{R}^{m+1}$. We also obtain an upper bound to the maximum height that a compact embedded translator in $\mathbb{R}^{3}$ can achieve. On the other hand, we study graphical perturbations of translators, showing that asymptotic graphical perturbations of a graph translator of revolution remain a hypersurface of revolution. Finally, we prove that compact translators that lie between two parallel planes inherit the symmetries of their boundaries curves.
Solitons to Mean Curvature Flow in the hyperbolic 3-space
本文研究双曲3-空间中平均曲率流的平移子与旋转子，证明了两类旋转平移子的存在唯一性，建立切触原理并分类常平均曲率情形，同时将螺旋旋转子结果推广至双曲空间。
$K^α$-translators of offset surfaces
本文研究了三维欧几里得空间中平行曲面与管状曲面上的 $K^α$-平移子，证明了管状曲面平移子必为旋转曲面，并探讨了特定旋转曲面及其平行曲面成为平移子的条件。
Maximum Principles and Consequences for $γ$-translators in $\mathbb{R}^{n+1}$
本文利用极大值原理研究一般外蕴曲率流中的 $γ$-平移子，证明切向原理、闭解与整解不存在性，并确立渐近于圆柱的严格凸单端解必为“碗”型解。
Maximum Principles and Consequences for $γ$-translators in $\mathbb{R}^{n+1}$ II
本文研究完全非线性外蕴曲率流平移孤立子，推广Spruck-Xiao与Spruck-Sun的凸性结果，并在曲率约束下刻画grim reaper柱面族。
Classification of noncollapsed translators in $\mathbb{R}^4$
本文分类了 $\mathbb{R}^4$ 及四维流形中三维超曲面平均曲率流的非坍塌奇点模型，证明其只能为三类已知平移超曲面之一。
Uniqueness of Semigraphical Translators
本文证明了$\mathbb{R}^3$中平均曲率流的音叉型和平板型平移子唯一性的猜想，采用弧计数论证与旋转极值原理，讨论了半图平移子分类及极限的应用，强化了相关紧性结果。
The class of grim reapers in $\mathbb{H}^2\times\mathbb{R}$
本文研究了乘积空间 $\mathbb{H}^2\times\mathbb{R}$ 中平均曲率流的平移子，对在单参数平移群下不变的 grim reaper 进行了完整分类，并描述了其几何性质与显式参数化。
We prove interior gradient estimate and second order estimate for the $Q_k$-flow and $Q_k$-translators in $\mathbb{R}^{n+1}$. In addition, we show that $Q_k$-translator which are asymptotic to $o(|x|)$ cannot exist.
Classification of Semigraphical Translators
本文基于Hoffman-Martín-White的工作，完成R^3中均曲率流半图形翻译器的分类，证明上半平面无解及pitchfork和helicoid翻译器的唯一性。
Böröczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, $L_p$-Minkowski and $L_p$-Brunn-Minkowski inequalities for two convex bodies in $\mathbb{R}^3$.
The linearized translator equation and applications
本文通过建立障碍估计和 Fredholm 理论，结合 Lyapunov-Schmidt 约化，证明了 $\mathbb{R}^4$ 中非塌缩平移子空间为有限维解析簇，并完成其分类，表明 Hoffman-Ilmanen-Martin-White 族由尖端最小主曲率唯一确定。
Enhanced profile estimates for ovals and translators
本文针对古代椭圆与非塌缩平移孤立子的轮廓函数，推导出尖锐的梯度估计和 Hessian 估计，分别增强了 Angenent-Daskalopoulos-Sesum 的 $C^0$-估计并作为其二次凹性估计的逆命题。
We show that the Bowl soliton in $\mathbb{R}^3$ is the unique translating solutions of the mean curvature flow which has the family of shrinking cylinders as an asymptotic shrinker at $-\infty$. As an application, we show that for a generic mean curvature flow, all (non-static) translating limit flows are the bowl soliton. The crucial point is that we do not make any global convexity assumption, while as the same time, the asymptotic requirement is very weak.
In this work we firstly classify space-like surfaces in Minkowski space $\mathbb E^4_1$, de-Sitter space $\mathbb S^3_1$ and hyperbolic space $\mathbb H^3$ with harmonic Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of the first kind. We also give some explicit examples.
The aim of this paper is to classify cohomogeneity one isometric actions on the 4-dimensional Minkowski space $\mathbb{R}^{3,1}$, up to orbit equivalence. Representations, up to conjugacy, of the acting groups in $O(3,1)\ltimes \mathbb{R}^{3,1}$ are given in both cases, proper and non-proper actions. When the action is proper, the orbits and the orbit spaces are determined.
In this paper, we study Lorentzian hypersurfaces in Minkowski 5-space with non-diagonalizable shape operator whose characteristic polinomial is $(t-k_1)^2(t-k_3)(t-k_4)$ or $(t-k_1)^3(t-k_4)$. We proved that in these cases, a hypersurface is biharmonic if and only if it is minimal.
In this paper, under suitable settings, we can obtain the existence and uniqueness of solutions to a class of Hessian quotient equations with Dirichlet boundary condition in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, which can be seen as a prescribed curvature problem and a continuous work of [12].
In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed $2$-th Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane $\mathscr{H}^{n}(1)\subset\mathbb{R}^{n+1}_{1}$ of center at origin and radius $1$, can be proven.
Classification of quadratically pinched translators in higher codimension
研究高余维二次收缩平移子，证明在Andrews-Baker阈值处余维约化，并在更强收缩条件下得到刚性分类：平移子为仿射平面或碗状孤子。
In this paper, we investigate the evolution of spacelike curves in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike Grim Reaper curve as time tends to infinity.
Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies $K$ in $\mathbb{R}^3$ whose boundary surface consists of an open surface $S$ with constant Gauss curvature (respectively, constant mean curvature) and $m$ planar compact discs $\bar{D_1},...,\bar{D_m},$ such that the Gauss map of $S$ is a homeomorphism onto $\mathbb{S}^2-\{p_j\}_{j=1}^m$ and $D_j\bot p_j,$ for all $j.$ We der…
In this paper, we study biconservative hypersurfaces in the four dimensional Minkowski space $\mathbb E^4_1$. We give the complete explicit classification of biconservative hypersurfaces with diagonalizable shape operator in $\mathbb E^4_1$.
By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for $n$-dimensional compact spacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geq n+2$. We develop a new suitable technique, based on an integral formula on compact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a family of extrinsic upper bounds for the first eigenvalue of the Laplace operator for a compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each on…
In this paper we give a classification of closed and connected Lie groups, up to conjugacy in $Iso({\mathbb{R}^3_1})$, acting by cohomogeneity one on the three dimensional Minkowski space $\mathbb{R}^3_1$ in both cases, proper and nonproper actions. Then we determine causal characters of the orbits.
In this paper, we prove some rigidity theorems for complete translating solitons. Assume that the $L^q$-norm of the trace-free second fundamental form is finite, for some $q\in\mathbb{R}$ and using a Sobolev inequality, we show that translator must be hyperspace. Our results can be considered as a generalization of \cite{Ma, WXZ16, Xin15}. We also investigate a vanishing property for translators which states that there are no nontrivial $L_f^p\ (p\geq2)$ weighted harmonic $1$-forms on ${M}$ if the $L^n$-norm of the second fundamental form is bounded.
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of hyperbolic plane of center at origin and prescribed radius, which actually cor…
In this paper, we consider the evolution of spacelike graphic curves defined over a piece of hyperbola $\mathscr{H}^{1}(1)$, of center at origin and radius $1$, in the $2$ dimensional Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic curves converge smoothly to a piece of hyperbola of center at origin and prescribed radius, which actually corresponds to a constant functio…
In this paper, we study entire translating solutions $u(x)$ to a mean curvature flow equation in Minkowski space. We show that if $Σ=\{(x, u(x))| x\in\mathbb{R}^n\}$ is a strictly spacelike hypersurface, then $Σ$ reduces to a strictly convex rank k soliton in $\mathbb{R}^{k, 1}$ (after splitting off trivial factors) whose "blowdown" converges to a multiple $λ\in(0, 1)$ of a positively homogeneous degree one convex function in $\mathbb{R}^k$. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbati…
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piec…
In this paper, we define three types of 2-Ruled hypersurfaces in the Minkowski 4-space $\mathbb{E}^4_1$. We obtain Gaussian and mean curvatures of the 2-ruled hypersurfaces of type-1 and type-2, and some characterizations about its minimality. We also deal with the first Laplace-Beltrami operators of these types of 2-Ruled hypersurfaces in $\mathbb{E}^4_1$. Moreover, the importance of this paper is that the definition of these surfaces by using the octonions in $\mathbb{E}^4_1$. Thus, this is a new idea and make the paper original. We give an example of 2-ruled hypersurface constructed by octo…
We prove that, in Minkowski space, if a spacelike, $(n-1)$-convex hypersurface $M$ with constant $σ_{n-1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an $\mathbb{R}^{n,1}$ rigid motion, $M$ splits as a product $M^{n-1}\times\mathbb{R}.$ We also construct nontrivial examples of strictly convex, spacelike hypersurface $M$ with constant $σ_{n-1}$ curvature and bounded principal curvatures.
Constant curvature curves in dual affine and dual Lorentz-Minkowski planes
本文研究了双仿射和双洛伦兹-闵可夫斯基平面中曲率恒为双常数的曲线性质，揭示了其几何特征和分类方法。
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of hyperbolic plane of center at origin and prescribed radius, which a…
Darboux transformations of spacelike curves in the Lorentz-Minkowski plane
该研究借助Penrose图共形紧化，考察Lorentz-Minkowski平面中类空曲线的Lorentz-Darboux变换，揭示其奇点与爆破等独特性质，并讨论类型变化曲线。
Consider the Lorentz-Minkowski $3$-space $\mathbb{L}^3$ with the metric $dx^2+dy^2-dz^2$ in canonical coordinates $(x,y,z)$. A surface in $\mathbb{L}^3$ is said to be separable if satisfies an equation of the form $f(x)+g(y)+h(z)=0$ for some smooth functions $f$, $g$ and $h$ defined in open intervals of the real line. In this article we classify all zero mean curvature surfaces of separable type, providing a method of construction of examples.
New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms
本文利用局部约束逆曲率流，为双曲空间和球面中的静态凸超曲面建立了一类新的精确加权Alexandrov-Fenchel不等式与Minkowski型不等式。
Time-like constant slope surfaces and space-like Bertrand curves in Minkowski 3-space
研究了德西特空间中类时间曲线的几何性质，并揭示了类空间伯兰德曲线与螺旋线之间的关系，以及类时间常斜面与类空间伯兰德曲线的联系。
The classical Minkowski problem in Minkowski space asks, for a positive function $ϕ$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $ϕ(η)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $η$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $μ$ on $\mathbb{H}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $K$ such that the area measure of $K$ is $μ$. In the present paper we look at an equivariant version of the problem: …
In this paper, we examine timelike loxodromes on three kinds of Lorentzian helicoidal surfaces in Minkowski $n$--space. First, we obtain the first order ordinary differential equations which determine timelike loxodromes on the Lorentzian helicoidal surfaces in $\mathbb{E}^n_1$ according to the causal characters of their meridian curves. Then, by finding general solutions, we get the explicit parametrizations of such timelike loxodromes. In particular, we investigate the timelike loxodromes on the three kinds of Lorentzian right helicoidal surfaces in $\mathbb{E}^n_1$. Finally, we give an exam…
In this paper, we study the Lorentzian minimal surfaces in the Minkowski space-time with finite type Gauss map. First, we obtain the classification of this type of surfaces with pointwise 1-type Gauss map. Then, we proved that there are no Lorentzian minimal surface in the Minkowski space-time with null 2-type Gauss map.
In this article, we get classification theorems for a Ricci soliton on the pseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ taking the potential vector field as the tangent component of the position vector of the pseudo-Riemannian hypersurface, denoted by $(M,g,{\bf x}^T,λ)$ in both Riemannian and Lorentzian settings. First, we obtain the necessary and sufficient condition that a pseudo-Riemannian hypersurface $(M,g)$ in $\mathbb{E}^4_1$ admits a Ricci soliton $(M,g,{\bf x}^T,λ)$. In each of the form of the shape operator of a pseudo-Riemannian hypersurface, we obtain cha…
A canonical normal null direction on a spacelike surface in the four dimensional Minkowski space $\mathbb{R}^{3,1}$ is a parallel vector field $Z$ on $\mathbb{R}^{3,1}$ such that the normal component of $Z$ on the surface is a lightlike vector field. We describe the geometric properties of a spacelike surface endowed with a canonical normal null direction and we obtain some characterizations of these surfaces. Moreover, using their Gauss map we study other properties of these surface: the associated ellipse of curvature and their asymptotic directions. Finally, we give two different ways to cr…
In this paper we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in $\mathbb{R}^{n+1}$ with speed $ψσ_k(λ)^α$, where $α$ is a positive constant, $σ_k(λ)$ is the $k$-th elementary symmetric polynomial of the principal radii of curvature and $ψ$ is a preassigned positive smooth function defined on $\mathbb{S}^n$. We prove that under some assumptions of $ψ$, the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known $L^p$ Christoffel-Minkowski problem $u^{1-p}( x ) σ_k \left( \nabla^2u+uI\right)=cψ(…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn-Minkowski inequality. A reverse isometric inequality for convex bodies is also given.
We introduce and study deformation $T_{{\bf b},ϕ}$ of Minkowski norms in $\mathbb{R}^n$, determined by a set ${\bf b}=(β_1,\ldots,β_p)$ of linearly independent 1-forms and a smooth positive function $ϕ$ of $p$ variables. In particular, the $T_{{\bf b},ϕ}$-image of a Euclidean norm $α$ is a Minkowski norm, whose indicatrix is a rotation hypersurface with a $p$-dimensional axis passing through the origin. For $p=1$, our deformation generalizes construction of $(α,β)$-norm; the last ones form a rich class of "computable" Minkowski norms and play an important role in Finsler geometry. We use compo…
Classification of ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia
本文研究Lorentz-Minkowski空间中关于原点惯性矩平稳的超曲面，在$\mathbb{L}^3$中分类平稳直纹面，证明平面是唯一平稳柱面，非柱面分类取决于直母线因果特征并给出显式参数化。
For a smooth strongly convex Minkowski norm $F:\mathbb{R}^n \to \mathbb{R}_{\geq0}$, we study isometries of the Hessian metric corresponding to the function $E=\tfrac12F^2$. Under the additional assumption that $F$ is invariant with respect to the standard action of $SO(k)\times SO(n-k)$, we prove a conjecture of Laugwitz stated in 1965. Further, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$-symmetry
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