# Parabolic Omori-Yau maximum principle for mean curvature flow and some applications

## TL;DR

This paper develops a parabolic Omori-Yau maximum principle for mean curvature flow in spaces with curvature bounds, and applies it to preserve Gauss map images and analyze self-shrinkers.

## Contribution

It introduces a new parabolic maximum principle for mean curvature flow and extends existing results to non-compact cases and self-shrinkers.

## Key findings

- Preservation of Gauss map image under mean curvature flow.
- Extension of Omori-Yau maximum principle to self-shrinkers.
- Generalization of Wang's result to non-compact immersions.

## Abstract

We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on $\ell$-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental forms. This generalizes the result of Wang \cite{Wang} for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in \cite{CJQ}.