Some variants of discrete positive mass theorems on graphs

TL;DR

This paper introduces the concept of asymptotically flat graphs, defines a discrete ADM mass, and proves a positive mass theorem for certain graph classes, revealing rigidity results related to curvature conditions.

Contribution

It formulates a discrete positive mass conjecture on graphs and proves the positive mass theorem for asymptotically flat grid-like graphs, establishing a discrete rigidity result.

Findings

Discrete ADM mass defined on asymptotically flat graphs

Positive mass theorem proved for grid-like graphs

Discrete rigidity result for non-negative Ricci curvature

Abstract

Inspired by asymptotically flat manifolds, we introduce the concept of asymptotically flat graphs and define the discrete ADM mass on them. We formulate the discrete positive mass conjecture based on the scalar curvature in the sense of Ollivier curvature, and prove the positive mass theorem for asymptotically flat graphs that are combinatorially isomorphic to grid graphs. As a corollary, the discrete torus does not admit positive scalar curvature. We prove a weaker version of the positive mass conjecture: an asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph. Hence the combinatorial structure of an asymptotically flat graph is determined by the curvature condition, which is a discrete analog of the rigidity part for the positive mass theorem. The key tool for the proof is the discrete harmonic function of linear growth associated with the salami structure.