Stability of $\mathbb{Z}^2$ configurations in 3D

TL;DR

This paper studies the stability and rigidity of finite point configurations in the integer lattice , focusing on angle-rigid structures and providing conditions for their stability against deformations.

Contribution

It introduces the concept of angle-rigidity for  configurations in 3D and establishes necessary and sufficient conditions for stability within certain subclasses.

Findings

Identifies conditions preventing angle-rigidity in  configurations.

Proves the sufficiency of these conditions for stability.

Establishes necessity of conditions in specific subclasses.

Abstract

Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two- and three-body contributions. Our main focus is on characterizing those configurations which cannot be deformed without changing distances between first neighbors or angles formed by pairs of first neighbors. Such configurations are called {\it angle-rigid}. We tackle this question in the class of finite configurations in $\mathbb{Z}^2$, seen as planar three-dimensional point sets. A sufficient condition preventing angle-rigidity is presented. This condition is also proved to be necessary when restricted to specific subclasses of configurations.