Nondegenerate neck pinches along the mean curvature flow

Abstract

We show that for generic smooth compact initial surfaces the mean curvature flow in $\mathbb{R}^3$ has spherical or nondegenerate neck pinch singularities at the first singular time. In particular the singularities at the first singular time are isolated in spacetime. As an application we give a new approach to constructing a mean curvature flow with surgery for smooth compact initial surfaces in $\mathbb{R}^3$.