The Mean Curvature Flow in Minkowski Spaces

TL;DR

This paper extends the concept of mean curvature flow to Minkowski spaces, establishing existence, uniqueness, and properties such as convexity preservation, with implications for geometry and physics.

Contribution

It introduces the mean curvature flow in Minkowski spaces, providing foundational results including existence, uniqueness, and convexity preservation.

Findings

Existence and uniqueness of solutions in Minkowski spaces

Flow preserves convexity and mean convexity

Derived comparison principles for the flow

Abstract

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we will prove the existence and uniqueness for solution of the mean curvature flow and prove that the flow preserves the convexity and mean convexity. We also derive some comparison principles for the mean curvature flow.