# Eventual Regularization of Fractional Mean Curvature Flow

## TL;DR

This paper proves that under fractional mean curvature flow, any open set initially close to a Lipschitz subgraph becomes a Lipschitz subgraph in finite time, demonstrating a novel regularizing effect for nonlocal curvature flows.

## Contribution

It establishes the first regularization result for weak solutions to nonlocal fractional mean curvature flow, highlighting a key difference from classical mean curvature flow.

## Key findings

- Finite time Lipschitz regularization for fractional mean curvature flow
- Quantitative and nonlocal proof technique
- First such result for weak solutions in nonlocal curvature flows

## Abstract

We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.