Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

Axioms

Benoît Loisel¹  Pascal Romon² 

[1] ENS de Lyon, 15 parvis René Descartes - BP 7000 69342 Lyon Cedex 07, France;Université Paris-Est, LAMA (UMR 8050), Marne-la-Vallée F-77454, France;

关键词: discrete curvature;    optimal transportation;    graph theory;    discrete Laplacian;    tiling;   

DOI  :  10.3390/axioms3010119

来源: DOAJ

【 摘 要 】

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu and Yau and Jost and Liu have used and extended this notion for graphs, giving estimates for the curvature and, hence, the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific, but crucial case of polyhedral surfaces.

【 授权许可】

Unknown