Related papers: Rigidity theorems for constant weighted mean curva…

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.…

Let $M^{n+1}$ be a closed manifold of dimension $3\le n+1\le 7$ equipped with a generic Riemannian metric $g$. Let $c$ be a positive number. We show that, either there exist infinitely many distinct closed hypersurfaces with constant mean…

As first noted in Korevaar, Kusner and Solomon ("KKS"), constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields.In Theorem 3.5 here, we generalize that law by relaxing the…

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 prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous…

$\lambda$-self-expanders $\Sigma$ in $\mathbb{R}^{n+1}$ are the solutions of the isoperimetric problem with respect to the same weighted area form as in the study of the self-expanders. In this paper, we mainly extend the results on…

In this paper, we present two rigidity results for stable constant mean curvature (CMC) surfaces immersed in $3$-manifolds with positive scalar curvature, assuming that the Hawking mass is zero. In the first result, we assume the surface to…

In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an…

Let $G$ be a compact connected subgroup of $SO(n+1)$. In $\mathbb{R}^{n+1}$, we gain interior $G$-symmetry for minimal hypersurfaces and hypersurfaces of constant mean curvature (CMC) which have $G$-invariant boundaries and $G$-invariant…

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 [16] there was proved that any biharmonic hypersurface with at most three distinct principal curvatures in space forms has constant mean curvature. At the very last step of the proof, the argument relied on the fact that the resultant of…

Let $M$ be an $n$-dimensional closed hypersurface with constant mean curvature and constant scalar curvature in an unit sphere. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form respectively. We…

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…

The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in $\mathbb{R}^4$, while they do not exist in positively curved closed…

In this paper we prove some general results on constant mean curvature lamination limits of certain sequences of compact surfaces $M_n$ embedded in $\mathbb R^3$ with constant mean curvature $H_n$ and fixed finite genus, when the boundaries…

In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally…

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of…

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted…

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 develop the compactness theorem for $\lambda$-surface in $\mathbb R^3$ with uniform $\lambda$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for…