Related papers: On mean curvature flow translators with prescribed…

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we…

We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional…

We study succinctness as a measure of the expressive power of transformers. Succinctness -- how compactly a formalism can describe a language relative to other formalisms -- is a classical notion in logic and automata theory. We prove that…

In this paper, we establish nonexistence results for complete translating solitons of the r-mean curvature flow under suitable growth conditions on the (r-1)-mean curvature and on the norm of the second fundamental form. We first show that…

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…

We prove interior gradient estimate and second order estimate for the $Q_k$-flow and $Q_k$-translators in $\mathbb{R}^{n+1}$. In addition, we show that $Q_k$-translator which are asymptotic to $o(|x|)$ cannot exist.

In this paper, we prove that if $M_t\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, is the $n$-dimensional closed embedded $\mathcal{F}-$stable solution to mean curvature flow with mean curvature of $M_t$ is uniformly bounded on $[0,T)$ for…

A contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$…

We construct closed, embedded, ancient mean curvature flows in each dimension $n\ge 2$ with the topology of $S^1 \times S^{n-1}$. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the…

In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R} \times…

An interval translation map (ITM) is a map $T \colon I \to I$ defined as a piecewise translation on a finite partition of an interval $I$ into $r \ge 2$ subintervals. Unlike classical interval exchange transformations (IETs), the images of…

In this short note we study Bernstein's type theorem of translating solitons whose images of their Gauss maps are contained in compact subsets in an open hemisphere of the standard $\mathbf{S}^n$ (see Theorem 1.1). As a special case we get…

For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution…

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…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this…

Let $C\subset\mathbb{R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb{R}^{n+1}$ that are asymptotic to $C$. As an application, we prove…

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. In…

We show that any complete, immersed self-expander to the inverse mean curvature flow, which has one end asymptotic to a cylinder, or has two ends asymptotic to two coaxial cylinders, must be rotationally symmetric.

We prove the following result: Let $(M,g_0)$ be a complete noncompact manifold of dimension $n\geq 12$ with isotropic curvature bounded below by a positive constant, with scalar curvature bounded above, and with injectivity radius bounded…

While it is well known from examples that no interesting `halfspace theorem' holds for properly immersed complete $n$-dimensional self-translating mean curvature flow solitons in Euclidean space $\mathbb{R}^{n+1}$, we show that they must…