Jose Torres Santaella
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.
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On the topology of translating solitons of the mean curvature flow
本文给出了中曲率流中存在平移孤子的分类结果和拓扑障碍。
In this paper, we prove that the infimum of the mean curvature is zero for a translating solitons of hypersurface in $\re^{n+k}$. We give some conditions under which a complete hypersurface translating soliton is stable. We show that if the norm of its mean curvature is less than one, then the weighted volume may have exponent growth. We also study the Dirichlet problem for graphic translating solitons in higher codimensions.
Bernstein theorem for translating solitons of hypersurfaces
本文证明了$\re^{n+1}$中超曲面平移孤子的单调性公式和Bernstein型结果，给出了平移孤子是超平面的条件.我们还证明了$R^{n+k}$中超曲面平移孤立子的间隙定理，即如果孤立子的第二基本型的$L^n$范数足够小，则它是一个超平面。
Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space
本文研究了Minkowski空间中中曲率流旋转对称平移孤子的存在性、唯一性和渐近行为。我们还研究了这类流的一般孤子的渐近性和严格凸性。
We study new examples of translating solitons of the mean curvature flow, especially in Minkowski space. We consider for this purpose manifolds admitting submersions and cohomegeneity one actions by isometries on suitable open subsets. This general setting also covers the classical Euclidean examples. As an application, we completely classify timelike, invariant translating solitons by rotations and boosts in Minkowski space.
Remarks on Bernstein theorems for translating solitons
本文研究了平移孤子$M$在$R^{n+1}$中的势函数性质以及欧氏球与$M$相交时的体积增长。给出了平移孤子的Bernstein定理成立的条件。我们还给出了一个简单的证明的伯恩斯坦定理由于Bjueri-De Giorgi-Miranda的大纲。
Gauss map of translating solitons of mean curvature flow
本文研究了在标准$\mathbf{S}^n$的开半球中，高斯映射的像包含在紧子集中的平移孤子的Bernstein型定理（见定理1.1）。作为特例，我们得到了$\mathbf{R}^{n+1}$中极小子流形的一个经典Bernstein型定理（见推论1.2）.
We address the asymptotic behavior of the $α$-Gauss curvature flow, for $α>1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as $t \to +\infty$, locally smoothly to a translating soliton which is uniquely determined by the asymptotic cylinder of the initial hypersurface.
Simply connected translating solitons contained in slabs
在这项工作中，我们表明，$2$维，简单连接，平移孤子的中曲率流嵌入在一个$\mathbb{R}^3$板的熵严格小于$3$必须是平均凸的，因此，由于J. Spruck和L。肖，是凸的。最近，这种$2$维凸平移孤子已被Hoffman，Ilmanen，Martín和White完全分类，直到环境等距，垂直平面，（倾斜）死神柱，$Δ$ -翼和碗translater。这些都包含在一个板，除了旋转对称碗translater。新的例子表明熵的界是必要的。
In this paper we prove that two-dimensional translating solitons in $\mathbb{R}^3$ with finite $L$-index are homeomorphic to a plane or a cylinder and that a two-dimensional self-expander with finite $L$-index and sub exponential weighted volume growth has finite topology. We also prove that translating solitons and self-expanders have finite topology, provided the bottom of the spectrum of the $L$-stability operator is bounded from below and their weighted volume have subexponential growth.
Half-space theorems for translating solitons of the r-mean curvature flow
本文研究了r-均曲率流中translating soliton的半空间定理，证明了在特定生长条件下，此类soliton无法完全包含在某些半空间或锥体中。
We obtain a rigidity result of symplectic translating solitons via the complex phase map. It indicates that we can remove the bounded second fundamental form assumption for symplectic translating solitons in [13].
Hamiltonian L-stability of Lagrangian Translating Solitons
本文计算了平移孤子的F-泛函的一阶和二阶变分公式，并研究了拉格朗日平移孤子的哈密顿L-稳定性。证明了任意拉格朗日平移孤子都是哈密顿L稳定的。
Lagrangian $L$-stability of Lagrangian Translating Solitons
本文证明了任意拉格朗日平移孤子都是拉格朗日$L$稳定的。
By using a new test function and the gradient estimate technique, we obtain a better Bernstein type result of translating solitons.
Remarks on topology of stable translating solitons
我们证明了任何完备的$f$ -稳定平移孤子$M$都不存在不断开$M$的余维圈.作为推论，任何二维完全$f$稳定平移孤子都有亏格零。
The main result of this paper is a convexity estimate for translating solitons of extrinsic geometric flows which evolve under a $1$-homogeneous concave function in the principal curvatures. In addition, we show examples of these hypersurfaces in $\mathbb{R}^{n+1}$ for particular functions.
Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications
针对完全非线性外蕴曲率流的平移孤子，发展旋转理论，获得碗型平移子的渐近展开，并引入符号颈框架构造和分类悬链面型平移子，进而证明严格凸整体图平移子的唯一性和有界图平移子的悬链面障碍不存在性。
We construct Lagrangian translating solitons by desingularizing the intersection points between Lagrangian Grim Reaper cylinders with the same phase using special Lagrangian Lawlor necks. The resulting Lagrangian translating solitons could have arbitrarily many ends and non-contractible loops.
Rigidity theorems of Lagrangian and symplectic translating solitons
利用精细点态估计，证明了拉格朗日与辛平移孤子的更优刚性定理。
The translating soliton equation
本文给出了平移孤子方程的解析解，并着重研究了平面凸区域中的狄利克雷问题。
Stability of translating solutions to mean curvature flow
证明了中曲率流旋转对称平移解的稳定性。对于初始数据，收敛到这样的孤子在无穷远的空间，我们得到收敛大的时间，孤子没有施加任何衰减率。
The H-flow translating solitons in R^3 and R^4
受Ilmanen对应的启发，我们给出了欧氏四维空间中中曲率流的非最小平移算子的Hoffman-Osserman Gauss映射问题的一个显式解.提出了一个关于不存在Jenkins-Serrin型单位速度图形翻译器的猜想。
We study translating soliton solutions to the flow by powers of the curvature of curves in the plane. We characterize these solitons as critical curves for functionals depending on the curvature. More precisely, translating solitons to the flow by powers of the curvature are shown to be generalized elastic curves. In particular, focusing on the curve shortening flow, we deduce a new variational characterization of the grim reaper curve.
Rigidity of spacelike translating solitons in pseudo-Euclidean space
本文研究了伪欧几里得空间$\mathbb{R}^{m+n}_{n}$中类空平移孤子刚性定理的参数形式和非参数形式。首先，利用仿射技巧和经典梯度估计对$\mathbb{R}^{m+n}_{n}$中的$m$维完全类空平移孤子进行了分类，证明了$\mathbb{R}^{m+n}_{n}$中的完全类空平移孤子是类空$m$ -平面.这一结果提供了\cite{C-Q}中完全类空平移孤子不存在性定理的另一种证明，以及\cite{X-H}中刚性定理的一种简单证明.其次，我们将\cite{X-Z}中的整类空拉格朗日平移孤子的刚性定理推广到具有一般余维的类空平移孤子。作为定理的直接应用，我们得到了关于高斯象的两个有趣的推论。
Entire spacelike translating solitons in Minkowski space
本文通过在有界凸域构造凸类空解并在无穷远指定边界数据，得到了Minkowski空间中多个整体光滑凸严格类空平移孤子。
Non-existence for self-translating solitons
本文建立了完整的，适当嵌入的，中曲率流自平移孤子$Σ^n\subseteq \mathbb{R}^{n+1}$的存在的几何障碍，推广以前已知的不存在的条件，如圆柱有界性。
Complete translating solitons to the mean curvature flow in $\mathbb{R}^3$ with nonnegative mean curvature
证明了中曲率流的任意完全浸没的双侧平均凸平移孤子$Σ\subset \mathbb{R}^3$都是凸的。作为推论，$\mathbb{R}^3$中的整体平均凸图形平移孤子是轴对称的“碗孤子”。我们还证明了如果$Σ$的中曲率在无穷远处趋于零，则$Σ$可以表示为一个完整的图，“碗孤子”也是如此.最后我们对定义在带状区域上的所有局部严格凸图形平移孤立子进行了分类。
The translating soliton of mean curvature flow
研究了平移孤子的一些基本问题：体积增长、广义最大值原理、高斯映射以及与高斯映射有关的函数，最后对第二基本型的平方模进行了逐点估计和积分估计.这些估计给出了高余维欧几里得空间中平移孤子的刚性定理。
Translating solitons to symplectic and Lagrangian mean curvature flows
本文构造了辛中曲率流的有限个爆破例子，研究了辛平移孤子的性质。证明了具有$\max |A|=1$的平移辛孤子的Kähler角$α$满足$\sup |α|>\fracπ{4}\frac{|T|}{|T|+1}$，其中$T$是表面平移的方向.
We prove that any complete immersed globally orientable uniformly 2-convex translating soliton $Σ\subset \mathbb{R}^{n+1}$ for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in $\mathbb{R}^{n+1},\, n\geq 3 $ is the axisymmetric "bowl soliton".
We characterize all ruled translating solitons in Minkowski 3-space. In contrast to the Euclidean space, we find ruled translating solitons that are not cylindrical. These surfaces appear when the vector field that defines the rulings, viewed as a curve, is a lightlike straight line. We also classify all cylindrical translating solitons, obtaining surfaces that can be considered as analogous to the grim reapers of Euclidean space, but also other surfaces which have no a counterpart in the Euclidean space.
Finite entropy translating solitons in slabs
本文研究了slab中具有有限熵和亏格的翻译溶iton，通过翼数计算熵并证明了特定条件下翻译叉形溶iton的唯一性。
Translating solitons $C^1-$asymptotic to two half-hyperplanes
证明了平行于${\bf e}_{n+1}$的超平面是平移孤子$C^1-$渐近于$\R^{n+1}$中垂直柱外两个半超平面的唯一例子。
We prove that if a translating soliton can be expressed as the sum of two curves and one of these curves is planar, then the other curve is also planar and consequently the surface must be a plane or a grim reaper. We also investigate translating solitons that can be locally written as the product of two functions of one variable. We extend the results in Lorentz-Minkowski space.
In this paper, we prove that the translating solitons of the mean curvature flow in $\mathbb{R}^4$ which arise as blow up limit of embedded, mean convex mean curvature flow must have $SO(2)$ symmetry.
Some examples of self-similar solutions and translating solitons for mean curvature flow
通过推广Joyce，Lee和Tsui的方法，我们构造了拉格朗日中曲率流的自相似解和平移孤子的新例子.这些例子包括拉格朗日角任意小的例子，如乔伊斯、李和徐的例子。
In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in ${\mathbb{R}^3}$. As application we study when the euclidean Gauss map has a harmonic argument and solve a general Cauchy problem in this class of surfaces.
We consider translating solitons to flows by positive powers $α$ of the Gaussian curvature -- called $K^α$-flows -- in Riemannian products $M\times\mathbb R.$ We prove that, when $M$ is the Euclidean space $\mathbb R^n,$ the sphere $\mathbb S^n,$ or one of the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m,$ there exist complete rotational translating solitons to $K^α$-flow in $M\times\mathbb R$ for certain values of $α.$
The purpose of these notes is to provide an introduction to those who want to learn more about translating solitons for the mean curvature flow in $\mathbb{R}^3$, particularly those which are complete graphs over domains in $\mathbb{R}^2$. In this paper we describe a full classification of complete translating graphs in $\mathbb{R}^3$.
Extending Translating Solitons in Semi-Riemannian Manifolds
本文回顾了半黎曼流形中平移孤子的一些一般性质和定理。此外，我们调查那些是不变的周围空间的等距的李群的行动，通过关注的行为接近奇异轨道（如果有的话）和无穷大。然后，我们提供了一些相关的例子。
The Dirichlet problem for the $α$-translating soliton equation on a strip
证明了定义在$\r^2$带上的$α$ -平移孤子方程的狄利克雷问题经典解的存在性.我们使用Perron方法，其中一个家庭的严峻的收割者作为障碍，解决狄利克雷问题时，边界数据是由两个副本的凸函数。
Translating solitons from semi-Riemannian foliations
我们回顾了半黎曼流形$(M,g)$和实直线的乘积中的（垂直）平移孤子的概念。主要地，我们把我们的注意力限制在那些是一个光滑函数的图形。在处理淹没时，我们给出了从基流形到全空间（或相反）的平移孤子的提升（或投影）准则。特别地，李群作用的余维1轨道所形成的流形会产生这样的孤子，直到求解一阶常微分方程。这给了我们明确的标准下，一个函数的图形是一个孤子，我们用它们来构建许多例子的孤子，新的和旧的，在一个统一的方式。
In this paper, we generalize a previous result to higher dimension. We prove that uniformly 3-convex translating solitons of mean curvature flow in $\mathbb{R}^{n+1}$ which arise as blow up limit of embedded, mean convex mean curvature flow must have $SO(n-1)$ symmetry.
On complete embedded translating solitons of the mean curvature flow that are of finite genus
本文通过沿 Costa-Hoffman-Meeks 曲面对3个 Grim 抛物面的并集进行去奇化，构造了具有3个端和任意有限亏格的平均曲率流完备嵌入平移孤立子。
Rigidity results and topology at infinity of translating solitons of the mean curvature flow
本文得到了欧几里得空间中中曲率流平移孤子无穷远拓扑的刚性结果和障碍。我们的方法依赖于f-极小超曲面的理论。
Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane
利用曲线缩短流（包括自收缩和自膨胀曲线或螺线）的某些解，构造和表征了复欧氏平面中中曲率流的平移孤子的许多新例子.在第二个维度上，他们概括了乔伊斯、李和徐的\cite{JLT}。在我们的家庭中最简单的（非平凡）的例子被刻画为唯一的（非全测地线）哈密顿定常拉格朗日平移孤子的中曲率流在复欧几里德平面。
Translating solitons of the mean curvature flow in the space $\mathbb{H}^2\times\mathbb{R}$
本文研究了乘积空间$\mathbb{H}^2\times\mathbb{R}$中浸没曲面的中曲率流的自平移孤子理论。我们将这一理论与密度流形的理论联系起来，并利用这一关系，将这些平移孤子看作共形度量空间中的极小曲面。这些表面的显式的例子构造，我们研究现有的旋转对称的例子的渐近行为。最后，我们证明了一些唯一性和不存在性定理。
Compact $λ$-translating solitons with boundary
具有密度矢量$\vec{v}$的$λ$平移孤子是欧几里得空间${\mathbb R}^3$中的表面$Σ$，其平均曲率$H$满足$2H=2λ+\langle N,\vec{v}\rangle$，其中$N$是$Σ$的高斯映射.本文从边界的角度研究了紧致$λ$平移孤子的形状。如果$Γ$是一条给定的闭曲线，我们推导出在什么条件下$λ$上存在一个具有边界$Γ$的紧凑$λ$ -平移孤子$Σ$，并给出了$Σ$的表面积与高度的关系。最后研究了$Σ$的形状与$Γ$的形状之间的关系，特别是给出了$Σ$继承其边界$Γ$对称性的条件.
Given a unit vector $\textbf{v}\in\mathbb{R}^3$ and $λ\in\mathbb{R}$, a translating $λ$-soliton is a surface in $\mathbb{R}^3$ whose mean curvature $H$ satisfies $H=\langle N,\textbf{v}\rangle+λ,\ |\textbf{v}|=1$, where $N$ is the Gauss map of the surface. In this paper, we extend the phenomenon of instability of Plateau-Rayleigh for translating $λ$-solitons of cylindrical type, proving that long pieces of these surfaces are unstable. We will provide explicit bounds on the length of these surfaces. It will be also proved that if a translating $λ$-soliton is a graph, then it is a minimizer of t…
Bi-Halfspace and Convex Hull Theorems for Translating Solitons
本文证明了欧几里得空间中完备自平移平均曲率流孤立子的双半空间定理，并利用距离函数与Omori-Yau极大值原理，分类了此类孤立子的凸包。
Translating solitons in Riemannian products
本文研究了环境黎曼流形中完全Killing向量场所产生的流的不变性孤子。当周围流形是黎曼积$(\mathbb{R} \times P, {\rm d}t^2+g_0)$且Killing场是$X=\partial_t$时，会出现特殊情况。类似于欧几里德环境中发生的情况，我们称之为平移孤子。我们看到，$\mathbb{R} \times P$中的平移孤子可以看作是加权体积泛函的极小子流形。此外，我们表明，这种孤子出现在一个自然的方式在$\mathbb{R} \times P$中的中曲率流的单调性公式的上下文中。当$g_0$是旋转不变的且截面曲率为非正时，我们可以刻画出所有的旋转不变平移孤子。此外，我们使用这些家庭的新的例子作为障碍，推导出几个不存在的结果。
The global solution of the minimal surface flow and translating surfaces
本文研究凸平面区域上的极小曲面流，在角度条件下证明平移孤立子的存在性，并在适当条件下证明全局解存在且收敛到某个平移子。
A Bernstein type theorem for a two dimensional translating soliton in ${\mathbb R}^4$ with finite total curvature
本文证明R^4中具有有限全曲率的二维完备平移孤立子必为平面，该结论是其主要结果的推论。
Rigidity and Non-existence Results of $λ$-Translating Solitons
本文研究$\lambda$-平移孤子，证明当$\lambda>0$且$\inf H>\lambda$时其体积至少指数增长，并给出图形孤子的不存在性与刚性定理。
In this paper, we consider a translating soliton for the inverse mean curvature flow given as a graph of a function on a domain in a unit sphere whose level sets give isoparametric foliation. First, we show that such function is given as a composition of an isoparametric function on the unit sphere and a function which is given as a solution of a certain ordinary differential equation. Further, we analyze the shape of the graphs of the solutions of the ordinary differential equation. This analysis leads to the classification of the shape of such translating solitons for the inverse mean curvat…
Volume growth, entropy and stability for translating solitons
研究了中曲率流中平移孤子的体积增长、熵和稳定性。首先，我们证明，每一个完整的适当沉浸翻译至少有线性体积增长。然后，利用Huisken的单调性公式，计算了碗孤子和死神的熵。在小熵的情况下，我们还给出了$\mathbf{R}^3$中平移算子的曲率估计.最后，我们估计了平移算子的稳定性算子$L$的谱，并给出了$L$ -稳定平移算子的刚性结果.
Minimum principles and a priori estimates for some translating soliton type problems
本文研究了两类推广平移孤子问题的中曲率型问题。第一个结果证明了这些问题的解具有唯一的内部临界点。利用这一唯一性结果，我们接着通过适当的$C^0$-函数的一些最小值原理，在L. E.佩恩
We deal with solitons of the mean curvature flow. The definition of \textit{translating solitons on a light-like direction} in Minkowski 3-space is introduced. Firstly, we classify those which are graphical, \textit{translation surfaces}, obtaining space-like and time-like, entire and not entire, complete and incomplete examples. Among them, all our time-like examples are incomplete. The second family consists of those which are invariant by a 1-dimensional subgroup of parabolic motions, i.e, with light-like axis. The classification result implies that all examples of this second family have s…
Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space
本文研究了超几何空间中均曲率流翻译孤立子的动力稳定性，通过构造旋转对称解并结合回避原理和最大原理，证明了超球面的稳定性。
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