Related papers: $K^\alpha$-translators of offset surfaces

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.…

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…

We complete the classification of semigraphical translators for mean curvature flow in $\mathbb{R}^3$ that was initiated by Hoffman-Mart\'in-White. Specifically, we show that there is no solution to the translator equation on the upper…

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three…

We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space $\mathbb{R}^{3}$. We determine all ruled surfaces and all relative…

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form…

We study the geodesic flow on the normal line congruence of a minimal surface in ${\Bbb{R}}^3$ induced by the neutral K\"ahler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is…

We present a characterization for the rotational soliton for the curve shortening flow (CSF) on the revolution surfaces of $\mathbb{R}^3$. Furthermore, we describe the behavior of such curves by showing that the two ends of each open curve…

The isotropic 3-space \mathbb{I}^{3} is a real affine 3-space endowed with the metric dx^{2}+dy^{2}. In this paper we describe Weingarten and linear Weingarten affine translation surfaces in \mathbb{I}^{3}. Further we classify the affine…

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichm\"uller geodesic flow. These surfaces are real algebraic curves with three real…

In this paper we adapt some techniques developed for K3 surfaces, to study the geometry of a family of projective varieties in $\Pl_K^2 \times \Pl_K^2 \times \Pl_K^2$ defined as the intersection of a form of degree $(2,2,2)$ and a form of…

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the…

We study asymptotic behavior of nonparametric hypersurfaces moving by $\alpha$ powers of Gauss curvature $\alpha > 1/n$. Our work generalizes the results of V. Oliker [Oli91] for $\alpha= 1$.

In this paper we make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in $\mathbb{R}^{3}$. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and…

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…

We present an intrinsic Klotz-Osserman type theorem for surfaces in terms of Codazzi operators. Additionally, utilizing Simons' formula, we investigate surfaces with parallel mean curvature with non-positive Gaussian curvature in product…

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of…

Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results…

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar…

Although the Transformer translation model (Vaswani et al., 2017) has achieved state-of-the-art performance in a variety of translation tasks, how to use document-level context to deal with discourse phenomena problematic for Transformer…