Compactness and convexity results for closed relativistic strings

TL;DR

This paper investigates properties of closed relativistic strings, including their closure under uniform convergence, convex collapsing behavior, and examples of weak Lipschitz evolution, extending existing mathematical theories.

Contribution

It extends previous results on the closure of relativistic strings and analyzes convex collapsing profiles, introducing new examples of string evolution.

Findings

Closure characterization of relativistic strings under uniform convergence

Convex planar strings collapse behavior similar to curvature flow

Example of weak Lipschitz evolution from a square with unexpected features

Abstract

We study various properties of closed relativistic strings. In particular, we characterize their closure under uniform convergence, extending a previous result by Y. Brenier on graph-like unbounded strings, and we discuss some related examples. Then we study the collapsing profile of convex planar strings which start with zero initial velocity, and we obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves. We conclude the paper with the discussion of an example of weak Lipschitz evolution starting from the square in the plane, which exhibits some unexpected features.