# Rigidity properties of the hypercube via Bakry-Emery curvature

## TL;DR

This paper establishes that the hypercube uniquely satisfies certain sharp geometric and spectral inequalities in discrete graph settings, using curvature-based methods.

## Contribution

It provides the first discrete analogues of Cheng's and Obata's rigidity theorems, characterizing hypercubes via curvature and combinatorial properties.

## Key findings

- Rigidity results for discrete Bonnet-Myers diameter bound
- Rigidity results for Lichnerowicz eigenvalue estimate
- Hypercube characterized as the unique graph satisfying these inequalities

## Abstract

We give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as well as new direct methods which translate curvature to combinatorial properties. Our results can be seen as first known discrete analogues of Cheng's and Obata's rigidity theorems.