Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part II

2014/08/28 by Erlend Grong, Grong, Erlend, Anton Thalmaier +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.DG

paper · pdf · doi:10.48550/arxiv.1408.6872

31 pages, Part 2 of 2. To appear in Mathematische Zeitschrift

openalex publication_date 2014/08/28 · arxiv created 2015/07/29 · arxiv updated 2015/07/30 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves
• Generalized Curvature Condition for Subelliptic Diffusion Processes

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