Neck Pinching Dynamics Under Mean Curvature Flow

TL;DR

This paper investigates how surfaces of revolution evolve under mean curvature flow, demonstrating that certain initial shapes develop a finite-time pinch point, with detailed analysis of the neck pinching process.

Contribution

It provides a detailed description of neck pinching behavior for surfaces of revolution under mean curvature flow, especially near cylindrical initial conditions.

Findings

Surfaces form a neck that pinches in finite time.

Pinching occurs at a single point.

Detailed dynamics of the neck pinch are characterized.

Abstract

In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.