# Convex ancient solutions to mean curvature flow

## TL;DR

This paper reviews and simplifies Wang's structure theory for convex ancient solutions to mean curvature flow, highlighting its implications and providing new rigidity results, thereby advancing understanding of these geometric flows.

## Contribution

It presents a simplified exposition of Wang's structure theory, introduces new rigidity results, and connects these findings to classifications in related curvature flows.

## Key findings

- Simplified analysis using monotonicity formula and Harnack inequality.
- New rigidity results for convex ancient solutions.
- Complete classification of convex ancient solutions in curve shortening flow.

## Abstract

X.-J. Wang proved a series of remarkable results on the structure of convex ancient solutions to mean curvature flow. Some of his results do not appear to be widely known, however, possibly due to the technical nature of his arguments and his exploitation of methods which are not widely used in mean curvature flow. In this expository article, we present Wang's structure theory and some of its consequences. We shall simplify some of Wang's analysis by making use of the monotonicity formula and the differential Harnack inequality, and obtain an important additional structure result by exploiting the latter. We conclude by showing that various rigidity results for convex ancient solutions and convex translators follow quite directly from the structure theory, including the new result of Corollary 8.3}. We recently provided a complete classification of convex ancient solutions to curve shortening flow by exploiting similar arguments.