On area-minimizing subgraphs in integer lattices

TL;DR

This paper introduces a new concept of area-minimizing subgraphs in infinite graphs, classifies them in 2D integer lattices, and explores their geometric properties in higher dimensions.

Contribution

It defines area-minimizing subgraphs using functions of bounded variations and provides a classification in 2D lattices along with geometric insights for higher dimensions.

Findings

Classification of area-minimizing subgraphs in 2D lattices

Geometric properties of these subgraphs in high dimensions

Extension of De Giorgi's functions of bounded variations to graph settings

Abstract

We introduce area-minimizing subgraphs in an infinite graph via the formulation of functions of bounded variations initiated by De Giorgi. We classify area-minimizing subgraphs in the two-dimensional integer lattice up to isomorphisms, and prove general geometric properties for those in high-dimensional cases.