Related papers: Rigidity and Curvature Estimates for Graphical Sel…

We give sharp bounds for the hyperbolic curvature of the level curve $|z|=|f(z)|$, when $f:\mathbb{D}\to\mathbb{D}$ is holomorphic on the unit disc $\mathbb{D}$ and $f(0)\neq0$, as well as for other related level curves. As a consequence,…

In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three dimensional Euclidean space is a round sphere, provided its mean curvature and the norm of its position vector have an upper…

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing…

In this paper, we prove that the infimum of the mean curvature is zero for a translating solitons of hypersurface in $\re^{n+k}$. We give some conditions under which a complete hypersurface translating soliton is stable. We show that if the…

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.

In this paper, we prove that any mean curvature flow translator $\Sigma^2 \subset \mathbb{R}^3$ with finite total curvature and one end must be a plane. We also prove that if the translator $\Sigma$ has multiple ends, they are asymptotic to…

In [5], Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinkers in $\mathbb{R}^{n+1}$, the entropy is uniquely minimized at the round sphere. They conjectured that, for $2\leq n\leq 6$, the round sphere…

In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…

A surface $\Sigma$ in the hyperbolic space $\h^3$ is called a horo-shrinker if its mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle$, where $(x,y,z)$ are the coordinates of $\h^3$ in the upper half-space model and $N$ is the unit…

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 study $\lambda$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_{\Sigma} e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems…

In this paper, we first introduce the concept of $\xi $-submanifold which is a natural generalization of self-shrinkers for the mean curvature flow and also an extension of $\lambda$-hypersurfaces to the higher codimension. Then, as the…

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant…

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 prove a suite of asymptotically sharp quadratic curvature pinching estimates for mean curvature flow in the sphere which generalize Simons' rigidity theorem for minimal hypersurfaces. We then obtain derivative estimates for the second…

In this paper, we generalize Colding and Minicozzi's work \cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the $F$-functional are derived and an…

A translating soliton is a hypersurface $M$ in $\mathbb{R}^{n+1}$ such that the family $M_t= M- t \,\mathbf{e}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at…

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or…

We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching…

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 $2g$…