# Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds

## TL;DR

This paper investigates high codimension mean curvature flow in Riemannian manifolds, establishing a codimension estimate and analyzing singularity formation, leading to convergence results towards codimension-one flows under certain pinching conditions.

## Contribution

It introduces a codimension estimate for high codimension flows and proves convergence to codimension-one flows at singularities under quadratic pinching.

## Key findings

- High curvature regions become approximately codimension one.
- Existence of rescaled flows converging to smooth codimension-one limits.
- Limiting flows are weakly convex and translate under cylindrical pinching.

## Abstract

We study the mean curvature flow of smooth $n$-dimensional compact submanifolds with quadratic pinching in a Riemannian manifold $\mathcal{N}^{n+m}$. Our main focus is on the case of high codimension, $m\geq 2$. We establish a codimension estimate that shows in regions of high curvature, the submanifold becomes approximately codimension one in a quantifiable way. This estimate enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space. Under a cylindrical type pinching, this limiting flow is weakly convex and moves by translation. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow.