Mathematische Zeitschrift

, Volume 282, Issue 1–2, pp 131–164 | Cite as

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part II

  • Erlend GrongEmail author
  • Anton Thalmaier


Using the curvature-dimension inequality proved in Part I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup \(P_t\) corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequality. These results require that the gradient of \(P_t f\) remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold.


Curvature-dimension inequality Sub-Riemannian geometry  Hypoelliptic operator Spectral gap Riemannian foliations 

Mathematics Subject Classification

58J35 53C17 58J99 



This work has been supported by the Fonds National de la Recherche Luxembourg (AFR 4736116 and OPEN Project GEOMREV).


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Mathematics Research Unit, FSTCUniversity of LuxembourgLuxembourgGrand Duchy of Luxembourg

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