Curvature Motion in Time-dependent Minkowski Planes

TL;DR

This paper investigates the evolution of curves under Minkowskian curvature flow in a time-dependent Minkowski plane, deriving formulas, existence conditions, and analyzing specific norm families, with implications for nonlinear PDEs.

Contribution

It introduces a framework for curvature flow in a dynamic Minkowski setting, including evolution formulas and existence criteria, extending classical results to time-dependent norms.

Findings

Derived evolution formulas for Minkowskian curvature flow.

Established conditions for short-time existence and convexity.

Connected the flow to nonlinear parabolic PDEs.

Abstract

In this paper we study a flow by minkowskian curvature where we have a different Minkowski plane at each time. We derive some evolution formulas, present sufficient hypotesis for the short time existence and convexity of solutions and study the motion considering a particular type of families of Minkowski norms. Also, as a corollary we establish a result about a certain family of nonlinear parabolic PDE's.