Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group

By Nicolas Juillet. 2009

146 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/b288846df2e212ce003607f051f45e03c60ea11d

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

New formulas for the Laplacian of distance functions and applications (2018, background): “Indeed the Heisenberg group H is an example of non-branching infinitesimally Hilbertian MCP(0, N) space [40] containing a line, which is homeomorphic and isomorphic as measure space to a splitting (indeed it is homeomorphic to R and the measure is exactly the n-dimensional Lebesgue measure) but it i”
Measure contraction properties of Carnot groups (2015, background): “Taking the limit for ε → 0+, we obtain that (G, d, μ) satisfies the MCP(0, N) with N < k + 3, but this is false by the previous step (these are the same arguments of [10]).”
Sub-Riemannian interpolation inequalities (2017, methodology): “For example, it is now well-known that the Heisenberg group equipped with a left-invariant measure, which is the simplest sub-Riemannian structure, does not satisfy any form of CD(K , N ), as proved in [33].”
Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality (2011, background): “However this criterion requires some ellipticity property from the diffusion operator and fails to hold even for simple subelliptic diffusion operators like the sub-Laplacian on the Heisenberg group (see [25]).”
The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds (2008, unknown): “In subriemannian geometry there is no real analogue of Ricci curvature; for instance, in Lott-Villani-Stürm sense (see [21], [28], [29]), the Ricci curvature of the simplest subelliptic model, the Heisenberg group, is −∞ (see [15]).”
Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds (2009, methodology): “D G ] 3 1 M ar”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, background): “X,F,m) satisﬁes the MCP(K,N) condition, see [Oht09]. • Sub-Riemannian manifolds. The following are all examples of essentially non-branching MCP(K,N)-spaces: the (2n + 1)-dimensional Heisenberg group [Jui09], any co-rank one Carnot group [Riz16], any ideal Carnot group [Rif13], any generalized H-typ”
On gradient bounds for the heat kernel on the Heisenberg group (2007, background): “For more details on the geometric aspects, we refer to [19, 33, 24].”
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, background): “However, as pointed out by Juillet in [34], the remarkable theory developed in these papers does not appear to be suited for sub-Riemannian manifolds.”