Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some variants of discrete positive mass theorems on graphs

Bobo Hua, Florentin Münch, Haohang Zhang

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.

Some variants of discrete positive mass theorems on graphs

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.
Paper Structure (15 sections, 16 theorems, 84 equations, 10 figures)

This paper contains 15 sections, 16 theorems, 84 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The positive mass theorem on manifolds
  3. The setting of weighted graphs
  4. Discrete positive mass conjecture and theorem on graphs
  5. The positive mass theorem on grid graphs
  6. Explicit formulas for the scalar curvature
  7. Proof of Theorem \ref{['thm:main1']}
  8. Some results and examples
  9. The positive mass theorem on general graphs
  10. Salami
  11. Proof of Theorem \ref{['thm:mainric']}
  12. The scalar curvature on discrete tori
  13. Discrete tori
  14. Other results
  15. Appendix

Key Result

Theorem 1.3

Suppose $(M^n,g)$ is an asymptotically flat manifold with $R(g)\ge 0$ and $n\le 7$. Then $E(g)\ge 0$ and $E(g)=0$ if and only if $(M,g)$ is isometric to $(\mathbb{R}^n,\delta),$ where $\delta$ is the Euclidean metric.

Figures (10)

  • Figure 1: Green edges indicate $S_r$ with $Q_r$ inside. Red edges indicate $E_r$ and blue edges indicate $\widetilde{E}_r.$
  • Figure 2: Illustration for the scalar curvature formula $R(x)$ when $n=2$. Red edges indicate the plus terms and blue edges indicate the minus terms. Green and red edges are involved in the absolute terms.
  • Figure 3: We first show that the weights for edges on lines outside $W$ are $1,$ such as the red and blue lines. Then every edge on them forces parallel edges to have the same weights. For example, the non-negative Ricci curvature of the green edge $e$ forces the pink edge $a$ to have the same weights with edge $b$ on the line above.
  • Figure 4: Illustration of $C_R$ and $h_1$ for $n=2$. The red line shows a shortest path connecting $L_{-R}$ and $L_R$ regardless of the structure of $K$.
  • Figure 5: The local structure of $K$ for $n=2$. The dashed lines represent edges to be determined. We will eliminate diagonal edges and keep only the horizontal and vertical edges.
  • ...and 5 more figures

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: The positive mass theorem
  • Definition 1.4
  • Definition 1.5: Discrete ADM mass
  • Conjecture 1.6: Discrete positive mass conjecture
  • Theorem 1.7: Positive mass theorem on grid graphs
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 26 more