Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator (Q1816479)

The classical theorem of Myers on diameter \([M]\) states that if \((M,g)\) is a complete, connected Riemannian manifold of dimension \(n\) \((\geq 2)\) such that \(\text{Ric}\geq(n-1)g\), then its diameter \(D=D(M)\) is less than or equal to \(\pi\). The authors prove an analogue of Myers's theorem for an abstract Markov generator and provide at the same time a new analytic proof of this result based on Sobolev inequalities. In particular, how to get exact bounds on the diameter in terms of the Sobolev constant is shown.