Related papers: The Structure of Translating Surfaces with Finite …

The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of…

We prove that any complete immersed two-sided mean convex translating soliton $\Sigma \subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in…

Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the…

In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most…

In this paper, we study the rigidity results of complete graphical translating hypersurfaces when the translating direction is not in the graphical direction. We proved that any entire graphical translating surface in the translating…

We prove that a vertical plane is the only complete translator, properly immersed in $\mathbb{R}^3$ and having finite topology, whose ends are asymptotic to vertical planes.

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…

Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along…

We study translating solitons for the mean curvature flow, $\Sigma^2\subseteq\mathbb{R}^3$ which are contained in slabs, and are of finite genus and finite entropy. As a first consequence of our results, we can enumerate connected…

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and…

We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.

We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.

In this work, we study complete properly immersed translators in the product space $\mathbb H^2\times\mathbb R$, focusing on their asymptotic behavior at infinity. We classify the asymptotic boundary components of these translators under…

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $\nu = \meta{N}{\partial_t}$ does not…

A \emph{surface of translation} is a sum $(u,v)\mapsto\gt\alpha(u)+\gt\beta(v)$ of two space curves: a \emph{path} $\gt\alpha$ and a \emph{profile} $\gt\beta$. A fundamental problem of differential geometry and shell theory is to determine…

We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit…

A $\lambda$-translating soliton with density vector $\vec{v}$ is a surface $\Sigma$ in Euclidean space ${\mathbb R}^3$ whose mean curvature $H$ satisfies $2H=2\lambda+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map of $\Sigma$. In…

In this paper, we prove a half-space theorem with respect to constant mean curvature $1/2$ entire graphs in $\mathbb{E(-1,\tau)}$. If $\Sigma$ is such an entire graph and $\Sigma'$ is a properly immersed constant mean curvature $1/2$…

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\mathbb{R}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha >\frac12$. We show that any such curve having in addition its…

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating…