相关论文: Rotators-Translators to Mean Curvature Flow in $\m…
We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal…
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…
In this paper we study the theory of self translating solitons of the mean curvature flow of immersed surfaces in the product space $\mathbb{H}^2\times\mathbb{R}$. We relate this theory to the one of manifolds with density, and exploit this…
We consider {translators} (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic $3$-space $\mathbb H^3$, providing existence and classification results. More specifically, we show the existence and…
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…
We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal…
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…
A spacelike surface in Minkowski space $\mathbb{R}_1^3$ is called a $K^\alpha$-translator of the flow by the powers of Gauss curvature if satisfies $K^\alpha= \langle N,\vec{v}\rangle$, $\alpha \neq 0$, where $K$ is the Gauss curvature, $N$…
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…
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…
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…
Rotation vectors, as defined for homeomorphisms of the torus that are isotopic to the identity, are generalized to such homeomorphisms of any complete Riemannian manifold with non-positive sectional curvature. These generalized rotation…
In this paper, we describe new annular examples of complete translating solitons for the mean curvature flow and how they are related to a family of translating graphs, the $\Delta$-wings. In addition, we will prove several related results…
In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…
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…
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 consider translating solitons to flows by positive powers $\alpha$ of the Gaussian curvature -- called $K^\alpha$-flows -- in Riemannian products $M\times\mathbb R.$ We prove that, when $M$ is the Euclidean space $\mathbb R^n,$ the…
In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in…
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…
The aim of this paper is to introduce a notion of mean curvature flow soliton general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces.