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International Mathematics Research Notices Advance Access published online on February 28, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp019
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© The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group

Nicolas Juillet

Institut Fourier BP 74, UMR 5582, Université Grenoble I, 38402 Saint-Martin-d'Hères Cedex, France, and Institute for Applied Mathematics, University of Bonn, Poppelsdorfer Allee 82, 53 115 Bonn, Germany

Correspondence: Correspondence to be sent to: nicolas.juillet{at}ujf-grenoble.fr

We prove that no curvature-dimension bound CD(K,N) holds in any Heisenberg group 13 n. On the contrary, the measure contraction property MCP(0, 2n+3) holds and is optimal for the dimension 2n+3. For the nonexistence of a curvature-dimension bound, we prove that the generalized "geodesic" Brunn–Minkowski inequality is false in 13 n. We also show in a new and direct way (and for all Formula ), that the general "multiplicative" Brunn–Minkowski inequality with dimension N>2n+1 is false.


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