Potential Analysis

, Volume 44, Issue 1, pp 27–41 | Cite as

On Measure Contraction Property without Ricci Curvature Lower Bound

  • Paul W. Y. Lee


Measure contraction properties M C P (K, N) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N, then M C P (K, N) is equivalent to Ricci curvature bounded below by K. On the other hand, it was observed in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013) that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy M C P (0,5). In this paper, we give sufficient conditions for a 2n+1 dimensional weakly Sasakian manifold to satisfy M C P (0, 2n + 3). This extends the above mentioned result on the Heisenberg group in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013).


Measure contraction property Ricci curvature lower bound Weakly Sasakian manifolds 

Mathematics Subject Classification (2010)

53C21 53C23 


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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.The Chinese University of Hong KongShatinHong Kong

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