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Comparison theorems on H-type sub-Riemannian manifolds.
Fabrice Baudoin1, Erlend Grong2, Luca Rizzi3,4,5
1Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark.
Calculus of Variations and Partial Differential Equations
|May 8, 2025
Summary
This study introduces uniform sub-Hessian and sub-Laplacian comparison theorems for H-type sub-Riemannian manifolds. It also presents a sharp sub-Riemannian Bonnet-Myers theorem applicable to a broader class of manifolds.
Area of Science:
- Differential Geometry
- Geometric Analysis
- Sub-Riemannian Geometry
Background:
- Sub-Riemannian geometry offers a framework for studying degenerate structures.
- Previous work established Bonnet-Myers theorems on specific contact manifolds.
Purpose of the Study:
- To establish uniform comparison theorems for approximating Riemannian metrics.
- To generalize the sub-Riemannian Bonnet-Myers theorem to H-type manifolds.
Main Methods:
- Utilizing a family of approximating Riemannian metrics.
- Developing sub-Hessian and sub-Laplacian comparison techniques.
- Extending existing theorems to a more general setting.
Main Results:
- Uniform sub-Hessian and sub-Laplacian comparison theorems established.
- A sharp sub-Riemannian Bonnet-Myers theorem proved for H-type manifolds.
- Generalization of prior results to a wider class of manifolds.
Conclusions:
- The findings provide essential tools for analyzing H-type sub-Riemannian manifolds.
- The generalized Bonnet-Myers theorem deepens the understanding of curvature in this context.
Keywords:
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