On asymptotic behavior for singularities of the powers of mean curvature flow

Abstract

Let M ⁿ be a smooth, compact manifold without boundary, and F ₀: M ⁿ → R ⁿ⁺¹ a smooth immersion which is convex. The one-parameter families F(·, t): M ⁿ × [0, T) → R ⁿ⁺¹ of hypersurfaces M ⁿ _t = F(·, t)(M ⁿ) satisfy an initial value problem dt/dF(·, t) = −H ^k(·, t)ν(·, t), F(·, 0) = F ₀(·), where H is the mean curvature and ν(·, t) is the outer unit normal at F(·, t), such that −H ν = $\overrightarrow H$ is the mean curvature vector, and k > 0 is a constant. This problem is called H ^k-flow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type I and Type II. It is proved that for Type I singularity, the limiting hypersurface satisfies an elliptic equation; for Type II singularity, the limiting hypersurface must be a translating soliton.