Every Salami has two ends
Abstract
A salami is a connected, locally finite, weighted graph with nonnegative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover show that every salami is recurrent and admits harmonic functions with constant gradient. The proofs are based on extremal Lipschitz extensions, a variational principle and the study of harmonic functions. Assuming a lower bound on the edge weight, we prove that salamis are quasiisometric to the line, that the space of all harmonic functions has finite dimension, and that the space of subexponentially growing harmonic functions is twodimensional. Moreover, we give a ChengYau gradient estimate for harmonic functions on balls.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 DOI:
 10.48550/arXiv.2105.11887
 arXiv:
 arXiv:2105.11887
 Bibcode:
 2021arXiv210511887H
 Keywords:

 Mathematics  Differential Geometry;
 53C21;
 53C23;
 05C75
 EPrint:
 26 pages, 3 figures