相关论文: Geometry of Two-dimensional Self-shrinkers

In Theorem 3.1 of [12], we proved a rigidity result for self-shrinkers under the integral condition on the norm of the second fundamental form. In this paper, we relax the such bound to any finite constant (see Theorem 4.4 for details).

In this paper, we establish the asymptotic expansion at infinity of gradient graph in dimension 2 with vanishing mean curvature at infinity. This corresponds to our previous results in higher dimensions and generalizes the results for…

In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology…

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 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.

In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example,…

By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\R^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in…

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space $\mathbb R^4$…

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 study space-like self-shrinkers of dimension $n$ in pseudo-Euclidean space $\ir{m+n}_m$with index $m$. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we…

In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker with polynomial volume growth in Euclidean space $\mathbb{R}^{n+1}$ is isometric to either $\mathbb{R}^{n}$,…

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends $\Sigma^n\subseteq\mathbb{R}^{n+1}$ that are hypersurfaces of revolution with circular boundaries.…

In this article we prove an unknottedness result for self shrinkers in $\mathbb{R}^3$ with multiple asymptotically conical ends which bound a handlebody in a natural sense, using the mean curvature flow. As a corollary of this and previous…

For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the…

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 paper, we generalize some halfspace type theorems for self-shrinkers of codimension 1 to the case of arbitrary codimension.

We study unknottedness for free boundary minimal surfaces in a three-dimensional Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary, and for self-shrinkers in the three-dimensional Euclidean space. For doing…

In this paper, we investigate classifications of $4$-dimensional simply connected complete noncompact nonflat shrinkers satisfying $Ric+\mathrm{Hess}\,f=\tfrac 12g$ with nonnegative Ricci curvature. One one hand, we show that if the…

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…

We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.