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Sub-Laplacian generalized curvature dimension inequalities on Riemannian foliations

Fabrice Baudoin, Guang Yang

TL;DR

This work develops a robust Bochner–Bakry–Émery framework for horizontal Laplacians on general Riemannian foliations without bundle-like or minimal-leaf assumptions. By employing an adapted metric connection, the authors derive explicit Bochner formulas for horizontal and vertical directions and introduce a curvature–dimension tensor $\mathfrak{R}$, enabling generalized CD inequalities that account for torsion and vertical mean curvature. The resulting inequalities yield Laplacian comparison theorems, Bonnet–Myers diameter bounds with explicit constants, stochastic completeness, eigenvalue estimates, and gradient/regularization bounds for the horizontal heat semigroup, with applicability to contact manifolds and Carnot groups of arbitrary step. The framework extends to broader non-foliated settings, offering a unified approach to sub-Riemannian-type analysis in structures lacking bundle-like metrics or minimal leaves, and providing tools for quantitative geometric and analytic conclusions in that broader class.

Abstract

We develop a Bochner theory and Bakry-Emery calculus for horizontal Laplacians associated with general Riemannian foliations. No bundle-like assumption on the metric, nor any total geodesicity or minimality condition on the leaves is imposed. Using a metric connection adapted to the horizontal-vertical splitting, we derive explicit Bochner formulas for the horizontal Laplacian acting on horizontal and vertical gradients, as well as a unified identity for the full gradient. These formulas involve horizontal Ricci curvature, torsion and vertical mean curvature terms intrinsic to the foliated structure. From these identities, we establish generalized curvature dimension inequalities, extending earlier results in sub-Riemannian geometry. As applications, we obtain horizontal Laplacian comparison theorems, Bonnet-Myers type compactness results with explicit diameter bounds, stochastic completeness, first eigenvalue estimates and gradient and regularization estimates for the horizontal heat semigroup. The framework applies, in particular, to contact manifolds and Carnot groups of arbitrary step.

Sub-Laplacian generalized curvature dimension inequalities on Riemannian foliations

TL;DR

This work develops a robust Bochner–Bakry–Émery framework for horizontal Laplacians on general Riemannian foliations without bundle-like or minimal-leaf assumptions. By employing an adapted metric connection, the authors derive explicit Bochner formulas for horizontal and vertical directions and introduce a curvature–dimension tensor , enabling generalized CD inequalities that account for torsion and vertical mean curvature. The resulting inequalities yield Laplacian comparison theorems, Bonnet–Myers diameter bounds with explicit constants, stochastic completeness, eigenvalue estimates, and gradient/regularization bounds for the horizontal heat semigroup, with applicability to contact manifolds and Carnot groups of arbitrary step. The framework extends to broader non-foliated settings, offering a unified approach to sub-Riemannian-type analysis in structures lacking bundle-like metrics or minimal leaves, and providing tools for quantitative geometric and analytic conclusions in that broader class.

Abstract

We develop a Bochner theory and Bakry-Emery calculus for horizontal Laplacians associated with general Riemannian foliations. No bundle-like assumption on the metric, nor any total geodesicity or minimality condition on the leaves is imposed. Using a metric connection adapted to the horizontal-vertical splitting, we derive explicit Bochner formulas for the horizontal Laplacian acting on horizontal and vertical gradients, as well as a unified identity for the full gradient. These formulas involve horizontal Ricci curvature, torsion and vertical mean curvature terms intrinsic to the foliated structure. From these identities, we establish generalized curvature dimension inequalities, extending earlier results in sub-Riemannian geometry. As applications, we obtain horizontal Laplacian comparison theorems, Bonnet-Myers type compactness results with explicit diameter bounds, stochastic completeness, first eigenvalue estimates and gradient and regularization estimates for the horizontal heat semigroup. The framework applies, in particular, to contact manifolds and Carnot groups of arbitrary step.
Paper Structure (23 sections, 19 theorems, 167 equations)

This paper contains 23 sections, 19 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Geometric setting
  3. Bochner's identities beyond the classical setting
  4. Generalized curvature dimension inequalities
  5. Applications
  6. Structure of the paper
  7. Preliminaries
  8. Setup and assumptions
  9. Horizontal Laplacian and heat kernel
  10. Horizontal Laplacian.
  11. Heat kernel.
  12. Fundamental connection
  13. Bochner's formulas and curvature dimension inequalities
  14. Horizontal and vertical Bochner's formulas for the horizontal Laplacian
  15. Bochner's formula for the horizontal Laplacian of the full gradient
  16. ...and 8 more sections

Key Result

Lemma 2.3

If $X_1,\cdots,X_n$ is a local orthonormal frame of horizontal vector fields then we locally have where $\mathrm H$ is the mean curvature vector field of the leaves.

Theorems & Definitions (42)

  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4: Hladky
  • Remark 2.5
  • Remark 2.6
  • Example 2.7: Contact manifold
  • Example 2.8: Carnot groups
  • Theorem 3.1
  • ...and 32 more