Related papers: Rigidity Results for Spacelike Self-Shrinkers via …
It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form…
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…
Using a maximum principle for self-shrinkers of the mean curvature flow, we give new proofs of a rigidity theorem for rotationally symmetric compact self-shrinkers and a result about the asymptotic behavior of self-shrinkers. This…
In this paper, we obtain a rigidity result of $2$-dimensional complete lagrangian self-shrinkers with constant squared norm $|\vec{H}|^{2}$ of the mean curvature vector in the Euclidean space $\mathbb{R}^{4}$. The same idea is also used to…
In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a classic geometric assumption, namely the union of all tangent affine submanifolds of a complete…
By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under…
We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem…
We study complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb{S}^{n+1}_{1}$ which is known as the steady state space $\mathcal{H}^{n+1}$. In this setting, under suitable constraints on the behavior of…
Let $M^{n}$ be an $n$-dimensional complete spacelike linear Weingarten submanifold immersed in a locally symmetric semi-Riemannian space $\mathbb{L}_{q}^{n+p}$ of index $q$, with parallel normalized mean curvature vector field and flat…
In this paper we analyze the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing…
In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity…
It is our purpose to study complete space-like self-expanders in the Minkovski space. By use of maximum principle of Omori-Yau type, we can obtain the rigidity theorems on $n$-dimensional complete space-like self-expanders in the Minkovski…
In this paper we present a rigidity theorem for locally isometric hypersurfaces with a curvature restriction in de Sitter space. This is an analogue to the case for Riemannian space forms given by Guan and Shen in [5].
We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on $\ell$-sectional curvature. We apply this to show that the image of Gauss map is preserved under a…
We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect…
In this paper, we prove gap results for complete self-shrinkers of the $r$-mean curvature flow involving a modified second fundamental form. These results extend previous results for self-shrinkers of the mean curvature flow due to Cao-Li…
By estimating the weighted volume, we obtain the optimal volume growth for Legendrian self-shrinkers. This, in turn, yields a rigidity theorem for entire smooth Legendrian self-shrinkers in the standard contact Euclidean (2n+1)-space.
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 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 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…