NDLI: Measure contraction properties of Carnot groups

Content Provider	SpringerLink
Author	Rizzi, Luca
Copyright Year	2016
Abstract	We prove that any corank 1 Carnot group of dimension $$k+1$$ equipped with a left-invariant measure satisfies the $$\mathrm {MCP}(K,N)$$ if and only if $$K \le 0$$ and $$N \ge k+3$$ . This generalizes the well known result by Juillet for the Heisenberg group $$\mathbb {H}_{k+1}$$ to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number $$k+3$$ coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent [the least N such that the $$\mathrm {MCP}(0,N)$$ is satisfied]. We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, our results improve a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat.
Ending Page	20
Page Count	20
Starting Page	1
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3
Volume Number	55
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-05-18
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc. Metric spaces, metrizability Sub-Riemannian geometry Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Geodesics Analysis Theoretical, Mathematical and Computational Physics Methods of Riemannian geometry, including PDE methods; curvature restrictions
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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