# A finiteness theorem via the mean curvature flow with surgery

## TL;DR

This paper employs the mean curvature flow with surgery to establish finiteness and isotopy results for convex hypersurfaces in Euclidean space, extending geometric analysis techniques.

## Contribution

It introduces a finiteness theorem for convex hypersurfaces using mean curvature flow with surgery, advancing understanding of their geometric classification.

## Key findings

- Proves a finiteness theorem for convex hypersurfaces
- Establishes isotopy existence results
- Extends Cheeger's compactness theorem to convex hypersurfaces

## Abstract

In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's compactness theorem) for the space of 2 convex hypersurfaces in $\mathbb{R}^{n+1}$.