Self-shrinkers with bounded |HA|

Smooth Compactness of Self-shrinkers

We prove a smooth compactness theorem for the space of embedded selfshrinkers in R. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all singularities and it plays an important role in studying generic mean curvature flow. 0. Introduction A surface Σ ⊂ R is said to be a self-shrinker if it satisfies (0.1) H ...

Uniqueness of Self-similar Shrinkers with Asymptotically Cylindrical Ends

In this paper, we show the uniqueness of smooth embedded selfshrinkers asymptotic to generalized cylinders of infinite order. Also, we construct non-rotationally symmetric self-shrinking ends asymptotic to generalized cylinders with rate as fast as any given polynomial.

Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends

Here H = div (n) is the mean curvature, n is the outward unit normal, x is the position vector and 〈, 〉 denotes the Euclidean inner product. One reason that selfshrinking solutions to the mean curvature flow are particularly interesting is that they provide singularity models of the flow; see [20, 21], [24] and [46]. Throughout, O is the origin of R; BR denotes the open ball in R n+1 centered a...

A Bernstein Type Theorem for Self-similar Shrinkers

In this note, we prove that smooth self-shrinkers in R, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is needed.

Some Bernstein Type Results of Graphical Self-Shrinkers with High Codimension in Euclidean Space