Equivalent Properties of CD Inequality on Graph
Yong Lin, Shuang Liu
TL;DR
This paper explores various equivalent properties of the curvature-dimension inequality on infinite graphs, including gradient estimates and Poincaré inequalities, and introduces a new curvature-dimension condition with similar equivalences.
Contribution
It establishes the equivalence of several properties related to $CD(n,K)$ and introduces a new $CDE'( abla, K)$ condition with similar properties on graphs.
Findings
Gradient estimate equivalence for $CD(n,K)$
Poincaré and reverse Poincaré inequalities equivalence
New curvature-dimension condition $CDE'( abla, K)$ with gradient estimate equivalence
Abstract
We study some equivalent properties of the curvature-dimension conditions inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincar\'e type inequalities and reverse Poincar\'e inequalities. And we also obtain one equivalent property of gradient estimate for a new notion of curvature-dimension conditions at the same assumption of graphs.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Geometry and complex manifolds