Related papers: On mean curvature flow translators with prescribed…

This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in the three-dimensional Euclidean space $\mathbb{R}^3.$ We prove that an immersed self-shrinker with finite $L$-index must be proper and of…

We consider a mean curvature flow in a cylinder with Robin boundary conditions, which can be used to model the interface motion in singular limit problems of the Allen-Cahn equation with nonlinear boundary conditions. It was shown in…

In this paper, we study a class of fully nonlinear contracting curvature flows of closed, uniformly convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with the normal speed $\Phi$ given by $r^\alpha F^\beta$ or $u^\alpha…

Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong…

We introduce a new version of expansiveness for flows. Let $M$ be a compact Riemannian manifold without boundary and $X$ be a $C^1$ vector field on $M$ that generates a flow $\varphi_t$ on $M$. We call $X$ {\it rescaling expansive} on a…

We provide a complete classification of groups that can be realized as isometry groups of a translation surface $M$ with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if $S$ has no…

This is the second of a three part study of relative free splitting complexes $\mathcal{FS}(\Gamma;\mathscr A)$, known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths $\tau_\phi \ge 0$ of the…

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p >…

Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the…

Motivated by Ilmanen's correspondence, we present an explicit solution to the prescribed Hoffman-Osserman Gauss map problem for non-minimal translators to the mean curvature flow in Euclidean 4-space. We propose a conjecture on the…

We study self-expanding solutions $M^m\subset\mathbb{R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and…

For an $H>0$ rotationally symmetric embedded torus $N_{0} \subset \mathbb{R}^{3}$, evolved by Inverse Mean Curvature Flow, we show that the total curvature $|A|$ remains bounded up to the singular time $T_{\max}$. We then show convergence…

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1)\times O(n)$ symmetry. We show they all have unique asymptotics as $t\to -\infty$ and we give precise asymptotic description…

A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact…

In this paper, we investigate the convexity of mean convex asymptotically conical self-expanders to the mean curvature flow in $\mathbb{R}^{n+1}$. Specifically, for $n\geq 3$, we show that any $n$-dimensional complete mean convex…

We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system $(X; T)$ embeds in the $K$-full shift with $h_{top}(T) < \log K $ and $\sharp Per_n(X; T) \leq…

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a…

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 this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with $|A|^2\le 1$ in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.

Let FI denote the category whose objects are the sets $[n] = \{1,\ldots, n\}$, and whose morphisms are injections. We study functors from the category FI into the category of sets. We write $\mathfrak{S}_n$ for the symmetric group on $[n]$.…