A Bernstein Type Theorem for Self-similar Shrinkers

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

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 ...

A Sharp Lower Bound for the Entropy of Closed Hypersurfaces up to Dimension Six

In [5], Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinkers in R, the entropy is uniquely minimized at the round sphere. They conjectured that, for 2 ≤ n ≤ 6, the round sphere minimizes the entropy among all closed smooth hypersurfaces. Using an appropriate weak mean curvature flow, we prove their conjecture. For these dimensions, our approach also gives...

The Morse index of mean curvature flow self - shrinkers by Zihan

In this thesis, we will introduce a notion of index of shrinkers of the mean curvature flow. We will then prove a gap theorem for the index of rotationally symmetric immersed shrinkers in R', namely, that such shrinkers have index at least 3, unless they are one of the stable ones: the sphere, the cylinder, or the plane. We also provide a generalization of the result to higher dimensions. Thesi...

On the approximation by Chlodowsky type generalization of (p,q)-Bernstein operators

In the present article, we introduce Chlodowsky variant of $(p,q)$-Bernstein operators and compute the moments for these operators which are used in proving our main results. Further, we study some approximation properties of these new operators, which include the rate of convergence using usual modulus of continuity and also the rate of convergence when the function $f$ belongs to the class Li...