Rigidity properties of the hypercube via Bakry-Emery curvature

Shiping Liu, Florentin Münch, Norbert Peyerimhoff

Published 2017-05-18Version 1

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.