相关论文: Translators of the mean curvature flow in the spec…

Given $\lambda\in\mathbb{R}$ and $\textbf{v}\in\mathbb{L}^3$, a $\lambda$-translator with velocity $\textbf{v}$ is an immersed surface in $\mathbb{L}^3$ whose mean curvature satisfies $H=\langle N,\textbf{v}\rangle+\lambda$, where $N$ is a…

We classify the translators to the mean curvature flow in the three-dimensional solvable group $Sol_3$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular we show that $Sol_3$…

A $\lambda$-translator is a surface in Euclidean space $\mathbb{R}^3$ whose Gauss curvature $K$ satisfies $K=\langle N, \vec{v} \rangle +\lambda$, where $N$ is the Gauss map, $\vec{v}$ is a fixed direction, and $\lambda \in \mathbb{R}$. In…

If $\xi$ is a Killing vector field of the hyperbolic space $\h^3$ whose flow are parabolic isometries, a surface $\Sigma\subset\h^3$ is a $\xi$-translator if its mean curvature $H$ satisfies $H=\langle N,\xi\rangle$, where $N$ is the unit…

In this paper, we consider translators (for the mean curvature flow) given by a graph of a function on a symmetric space $G/K$ of compact type which is invariant under a hyperpolar action on $G/K$. First, in the case of $G/K=SO(n+1)/SO(n)$,…

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. The problem is solved…

Translators can be regarded as submanifolds which satisfy the mean curvature flow equation when evolving by translations along a distinguished vector field of the ambient space. We study translators in Generalised Robertson-Walker…

In this study, we deal with non-degenerate translators of the mean curvature flow in the well-known hyperbolic Einstein's static universe. We classify translators foliated by horospheres and rotationally invariant ones, both space-like and…

A $\lambda$-translating soliton with density vector $\vec{v}$ is a surface in Euclidean space whose mean curvature $H$ satisfies $2H=2\lambda+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map. We classify all $\lambda$-translating…

A translating soliton is a hypersurface $M$ in $\mathbb{R}^{n+1}$ such that the family $M_t= M- t \,\mathbf{e}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at…

A $K^\alpha$-translator is a surface in Euclidean space $\r^3$ that moves by translations in a spatial direction and under the $K^\alpha$-flow, where $K$ is the Gauss curvature and $\alpha$ is a constant. We classify all…

In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the…

A soliton of the mean curvature flow in the product space $\mathbb{s}^2\times\mathbb{R}$ as a surface whose mean curvature $H$ satisfies the equation $H=\langle N,X\rangle$, where $N$ is the unit normal of the surface and $X$ is a Killing…

There exist four non-equivalent types of the translation hypersurfaces in the 4-dimensional isotropic space $\mathbb{I}^{4}$ generated by translating the curves lying in perpendicular $k-$planes $\left(k=2,3\right)$, due to its absolute…

We prove, in all dimensions $n\geq 2$, that there exists a convex translator lying in a slab of width $\pi\sec\theta$ in $\mathbb{R}^{n+1}$ (and in no smaller slab) if and only if $\theta\in[0,\frac{\pi}{2}]$. We also obtain convexity and…

We study translators of the mean curvature flow in the product space $\h^2\times\r$. In $\h^2\times\r$ there are three types of translations: vertical translations due to the factor $\r$ and parabolic and hyperbolic translations from…

A translation surface in the Heisenberg group is constructed as the product of two planar curves. We classify a type of such surfaces with vanishing intrinsic curvature by analyzing the determinant of their Gauss map

We construct a family of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as $z\to -\infty$…

We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to $0$ to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are…

We establish the existence of one-parameter families of helicoidal surfaces of $\mathbb H^2\times\mathbb R$ which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.