Every Salami has two ends

• Published in 2021
• Added on 2021-05-26

A salami is a connected, locally finite, weighted graph with non-negative 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 quasi-isometric to the line, that the space of all harmonic functions has finite dimension, and that the space of subexponentially growing harmonic functions is two-dimensional. Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on balls.

Links

• http://arxiv.org/abs/2105.11887v1
• http://arxiv.org/pdf/2105.11887v1

Other information

BibTeX entry

@article{EverySalamihastwoends,
	key = {EverySalamihastwoends},
	type = {article},
	title = {Every Salami has two ends},
	author = {Bobo Hua and Florentin M{\"{u}}nch},
	abstract = {A salami is a connected, locally finite, weighted graph with non-negative
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 quasi-isometric to the
line, that the space of all harmonic functions has finite dimension, and that
the space of subexponentially growing harmonic functions is two-dimensional.
Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on
balls.},
	comment = {},
	date_added = {2021-05-26},
	date_published = {2021-12-07},
	urls = {http://arxiv.org/abs/2105.11887v1,http://arxiv.org/pdf/2105.11887v1},
	collections = {attention-grabbing-titles,food},
	url = {http://arxiv.org/abs/2105.11887v1 http://arxiv.org/pdf/2105.11887v1},
	year = 2021,
	urldate = {2021-05-26},
	archivePrefix = {arXiv},
	eprint = {2105.11887},
	primaryClass = {math.DG}
}