Siao-Hao Guo

If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow.

Given a mean curvature flow of compact, embedded $C^2$ surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.

J.J.L. Vel$\acute{a}$zquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type $\mathrm{II}$ singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the $C^0$ sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type $\mathrm{II}$ singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.

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 show that there is a $f$ self-shrinker (i.e. a hypersurface $\Sigma$ in ${\mathbb{R}}^{n+1}$ which satisfies $f\left({\kappa }_1,\cdots ,{\kappa }_n\right)+\frac{1}{2}X\cdot N=0$, where $X$ is the position vector, $N$ is the unit normal vector, and ${\kappa }_1,\cdots ,{\kappa }_n$ are principal curvatures of $\Sigma$) that is asymptotic to $\mathcal{C}$ at infinity.

Given a smooth, symmetric, homogeneous of degree one function $f=f\left({\lambda }_1,\cdots ,{\lambda }_n\right)$ satisfying ${\partial }_{i}f>0$ for all $i=1,\cdots ,n$, and an oriented, properly embedded smooth cone ${\mathcal{C}}^n$ in ${\mathbb{R}}^{n+1}$, we show that under some suitable conditions on $f$ and the covariant derivatives of the second fundamental form of $\mathcal{C}$, there is at most one $f$ self-shrinker (i.e. an oriented hypersurface ${\Sigma }^n$ in ${\mathbb{R}}^{n+1}$ for which $f\left({\kappa }_1,\cdots ,{\kappa }_n\right)+\frac{1}{2}X\cdot N=0$ holds, where $X$ is the position vector, $N$ is the unit normal vector, and ${\kappa }_1,\cdots ,{\kappa }_n$ are principal curvatures of $\Sigma$) that is asymptotic to the given cone $\mathcal{C}$ at infinity.