Related papers: Stability of translating solutions to mean curvatu…

In this paper, we consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a…

There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations.…

Analogous to the bowl soliton of mean curvature flow, we construct rotationally symmetric translating solutions to a very large class of extrinsic curvature flows, namely those whose speeds are $\alpha$-homogeneous ($\alpha>0$), elliptic…

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…

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity…

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then…

We study a generalized mean curvature flow involving a positive power of the mean curvature and a driving force. In this paper, we first construct all kinds of radially symmetric translating solutions, and then select one of them to satisfy…

In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting $C^{1,1}-$close to a strictly stable critical set of the perimeter $E$, exist for all times and…

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…

In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus $\mathbb T^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set…

We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore…

In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of…

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…

In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of…

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…

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…

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.

We study some basic problems of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps, finally we carry out point-wise estimates and integral estimates for the…

In this paper, we study entire translating solutions $u(x)$ to a mean curvature flow equation in Minkowski space. We show that if $\Sigma=\{(x, u(x))| x\in\mathbb{R}^n\}$ is a strictly spacelike hypersurface, then $\Sigma$ reduces to a…