# Nonconvex Surfaces which Flow to Round Points

## TL;DR

This paper extends Huisken's theorem by constructing various mean curvature flow examples, including pathological cases, demonstrating complex behaviors of surfaces shrinking to round points.

## Contribution

It introduces new classes of hypersurfaces that flow to round points, including pathological examples with unusual convergence properties.

## Key findings

- Constructed hypersurfaces that shrink to round points with no Gromov-Hausdorff convergence.
- Created flows where initial surfaces converge to space-filling surfaces.
- Produced surfaces of large area close to spheres that still shrink to round points.

## Abstract

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that exist on a uniform time interval, have uniformly bounded diameter, and shrink to round points, yet the sequence of initial surfaces has no subsequence converging in the Gromov-Hausdorff sense. Moreover, we find a sequence of flows which all shrink to round points, yet the initial surfaces converge to a space-filling surface. Also constructed are surfaces of arbitrarily large area which are close in Hausdorff distance to the round sphere yet shrink to round points.