[Paper Review] Bishop and Laplacian comparison theorems on Sasakian manifolds

Paul W. Y. Lee, Chengbo Li|arXiv (Cornell University)|Oct 20, 2013

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper establishes Bishop and Laplacian comparison theorems for sub-Riemannian manifolds equipped with a natural structure on Sasakian manifolds. By analyzing the Tanaka-Webster curvature and using canonical frames and Jacobi field techniques, the authors derive volume and Laplacian comparison estimates that generalize prior 3D results, with equality conditions characterizing model spaces like the Heisenberg group.

ABSTRACT

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for a natural sub-Riemannian structure defined on Sasakian manifolds. This generalizes the earlier work for the three dimensional case.

Motivation & Objective

• To extend Bishop and Laplacian comparison theorems from 3D Sasakian manifolds to general 2n+1-dimensional Sasakian manifolds.
• To establish volume comparison theorems under lower bounds on specific components of the Tanaka-Webster curvature tensor.
• To derive a sub-Laplacian comparison estimate using curvature bounds on the Tanaka-Webster curvature, generalizing earlier results in the literature.
• To characterize equality cases in the comparison theorems, identifying when the manifold locally matches the geometry of the Heisenberg group or the complex Hopf fibration.

Proposed method

• The authors use the canonical frame construction from [8] and the parallel adapted frame formalism from [7] to simplify curvature computations on the Sasakian manifold.
• They define the sub-Riemannian distance and volume via the Riemannian volume form restricted to the contact distribution D = ker α₀.
• The proof relies on analyzing Jacobi fields along sub-Riemannian geodesics using the curvature of the Jacobi curve in the Lagrangian Grassmannian.
• Key curvature conditions are imposed on the Tanaka-Webster curvature tensor: non-negativity of two specific quadratic forms involving Jv and orthogonal vectors wᵢ.
• The Laplacian comparison theorem is derived via a Hessian comparison principle, with the comparison function h(r,z) depending on curvature bounds k₁ and k₂.
• The comparison functions h(r,z) are explicitly constructed using trigonometric and hyperbolic functions based on the signs of k₁ and k₂.

Experimental results

Research questions

• RQ1 Under what curvature conditions on the Tanaka-Webster curvature does the sub-Riemannian volume of a ball in a Sasakian manifold not exceed that of the Heisenberg group?
• RQ2 When does equality hold in the Bishop-type volume comparison theorem for Sasakian manifolds?
• RQ3 How can a Laplacian comparison theorem be formulated for the sub-Laplacian on Sasakian manifolds under curvature bounds?
• RQ4 What is the precise form of the comparison function h(r,z) that dominates the sub-Laplacian of the sub-Riemannian distance function?
• RQ5 How do the curvature conditions on the Tanaka-Webster curvature relate to the geometry of model spaces like the Heisenberg group and complex Hopf fibration?

Key findings

• The volume of a sub-Riemannian ball Bₓ(R) in a Sasakian manifold is bounded above by the volume of the corresponding ball in the Heisenberg group if the Tanaka-Webster curvature satisfies ∑⟨Rm*(wᵢ,v)v,wᵢ⟩ ≥ (2n−2)|v|² and ⟨Rm*(Jv,v)v,Jv⟩ ≥ 4|v|⁴.
• Equality in the volume comparison holds if and only if the curvature conditions are saturated: ⟨Rm*(Jv,v)v,Jv⟩ = 4|v|⁴ and ∑⟨Rm*(wᵢ,v)v,wᵢ⟩ = (2n−2)|v|² on the ball.
• The sub-Laplacian of the sub-Riemannian distance function Δₕd is bounded above by a comparison function h(r,z) that depends on curvature bounds k₁ and k₂, with explicit expressions in terms of trigonometric and hyperbolic functions.
• The comparison function h(r,z) is defined piecewise based on the signs of k₁ and k₂, incorporating terms involving √k₁, √k₂, cot, coth, sin, cos, sinh, and cosh.
• When k₁ ≥ 0 and k₂ ≥ 0, the comparison function involves sin and cos terms; when k₁ ≤ 0 or k₂ ≤ 0, it uses sinh and cosh, with appropriate sign adjustments.
• The equality case in the Laplacian comparison theorem occurs precisely when the curvature bounds are saturated: ⟨Rm*(Jv,v)v,Jv⟩ = k₁|v|⁴ and ∑⟨Rm*(wᵢ,v)v,wᵢ⟩ = (2n−2)k₂|v|².