# Entropy and generic mean curvature flow in curved ambient spaces

## TL;DR

This paper extends recent results on generic singularities of mean curvature flow to curved ambient spaces, using a weighted monotonicity formula to analyze entropy behavior and its continuity.

## Contribution

It introduces a method to analyze entropy and singularities of mean curvature flow in curved ambient spaces, generalizing previous Euclidean results.

## Key findings

- Derived an almost monotonicity of entropy in curved spaces
- Studied the continuity properties of the entropy functional
- Extended singularity analysis techniques to curved ambient spaces

## Abstract

In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity" for the entropy upon embedding into $\R^\ell$. We are also lead to study the continuity of the entropy functional in certain cases.