Related papers: Self-shrinkers with a rotational symmetry
Given a smooth, symmetric, homogeneous of degree one function $f\left(\lambda_{1},\cdots,\,\lambda_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\,n$, and a rotationally symmetric cone $\mathcal{C}$ in $\mathbb{R}^{n+1}$, we…
In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in $\mathbb{R}^{n+1}$. First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the…
We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact…
In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P $ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma$ contained in one of the closed…
We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in $\mathbb R^3$ that are not rotationally…
We construct many closed, embedded mean curvature self-shrinking surfaces $\Sigma_g^2\subseteq\mathbb{R}^3$ of high genus $g=2k$, $k\in \mathbb{N}$. Each of these shrinking solitons has isometry group equal to the dihedral group on $2g$…
In this article we show the existence of closed embedded self-shrinkers in $\Bbb{R}^{n+1}$ that are topologically of type $S^1\times M$, where $M\subset S^n$ is any isoparametric hypersurface in $S^n$ for which the multiplicities of the…
In this paper we show the existence of a closed, embedded $\lambda$-hypersurfaces $\Sigma \subset \mathbb{R}^{2n}$. The hypersurface is diffeomorhic to $\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$ and exhibits $SO(n)…
In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in $R^{n+1}$ an isoparametric cone if C is the cone over a…
Given a smooth, symmetric, homogeneous of degree one function $f=f\left(\lambda_{1},\cdots,\,\lambda_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\, n$, and an oriented, properly embedded smooth cone $\mathcal{C}^n$ in…
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…
We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.
As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo \cite{Guo} proved…
In R^3, let M be the infinite union of unit spheres whose centers lie at even integers on the x-axis; every pair of consecutive spheres touches at (2m+1, 0, 0). Desingularizing these point contacts yields Delaunay's classical constant mean…
In this paper, we investigate closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by…
Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show…
In this work we derive explicit entropy bounds for two classes of closed self-shrinkers: the class of embedded closed self-shrinkers recently constructed in arXiv:2207.04851 using isoparametric foliations of spheres, and the class of…
In this article, we study hypersurfaces $\Sigma\subset \mathbb{R}^{n+1}$ with constant weighted mean curvature. Recently, Wei-Peng proved a rigidity theorem for CWMC hypersurfaces that generalizes Le-Sesum classification theorem for…
In the paper, we construct, for $\lambda>0$, complete embedded and non-convex $\lambda$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $\lambda$-hypersurfaces share a common conclusion on the planar…
All complete, axially symmetric surfaces of constant mean curvature in R^3 lie in the one-parameter family D_tau of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond…