[Paper Review] Comparison theorems on H-type sub-Riemannian manifolds

Fabrice Baudoin, Erlend Grong+2 more|arXiv (Cornell University)|Sep 8, 2019

Geometric Analysis and Curvature Flows40 references4 citations

TL;DR

This paper establishes uniform sub-Hessian and sub-Laplacian comparison theorems for H-type sub-Riemannian manifolds by analyzing the limit of a canonical variation of Riemannian metrics. It proves a sharp sub-Riemannian Bonnet-Myers theorem that extends prior results on contact and quaternionic contact manifolds, showing that curvature bounds yield finite diameter estimates even in the sub-Riemannian limit.

ABSTRACT

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.

Motivation & Objective

• To develop uniform comparison estimates for the horizontal Laplacian and Hessian on H-type sub-Riemannian manifolds across a family of approximating Riemannian metrics.
• To overcome the failure of classical Riemannian comparison theorems in the sub-Riemannian limit due to unbounded curvature in the canonical variation.
• To prove a sharp sub-Riemannian Bonnet-Myers theorem valid for general H-type structures, extending known results on contact and 3-Sasakian manifolds.
• To establish that the sub-Riemannian limit of Riemannian comparison estimates yields finite diameter bounds under curvature conditions.
• To remove the $J^2$ condition in certain settings while preserving the validity of comparison theorems via geometric analysis of the Hladky connection and canonical variation.

Proposed method

• Utilizes the canonical variation of Riemannian metrics $g_\varepsilon = g_\mathcal{H} \oplus \frac{1}{\varepsilon}g_\mathcal{V}$, which converges to the sub-Riemannian metric as $\varepsilon \to 0$.
• Applies the Hladky connection and its associated curvature and torsion to define a metric-compatible connection with skew-symmetric torsion for comparison purposes.
• Derives a uniform comparison theorem for the horizontal Laplacian $\Delta_\mathcal{H} r_\varepsilon \leq F_\varepsilon(r_\varepsilon)$ valid for all $\varepsilon > 0$, with $F_\varepsilon$ converging to a finite limit as $\varepsilon \to 0$.
• Employs the index lemma and Jacobi field analysis on geodesics to compare the Hessian of the distance function under the connection $D$ with curvature and torsion terms.
• Uses the Hessian formula $\mathrm{Hess}^D(r)(X,X) = \int_0^r \left( \langle D_{\dot\gamma}V, \hat{D}_{\dot\gamma}V \rangle - \langle R(V,\dot\gamma)\dot\gamma, V \rangle \right) dt$ to relate curvature bounds to comparison estimates.
• Applies the comparison principle via the index form $I(W,W)$ and proves that the Hessian is bounded above by the index form of any vector field $W$ perpendicular to the geodesic.

Experimental results

Research questions

• RQ1 Can uniform sub-Hessian and sub-Laplacian comparison theorems be established for H-type sub-Riemannian manifolds across the canonical variation of Riemannian metrics?
• RQ2 Does the classical Laplacian comparison theorem fail in the sub-Riemannian limit due to curvature blow-up in the approximating metrics, and if so, can a uniform replacement be constructed?
• RQ3 Can a sharp sub-Riemannian Bonnet-Myers theorem be proven for general H-type manifolds without assuming the $J^2$ condition?
• RQ4 Is the limit of the Riemannian comparison estimates as $\varepsilon \to 0$ finite and meaningful in the sub-Riemannian setting?
• RQ5 How do curvature bounds on the H-type structure translate into diameter bounds in the sub-Riemannian limit?

Key findings

• The paper establishes a uniform sub-Laplacian comparison theorem of the form $\Delta_\mathcal{H} r_\varepsilon \leq \frac{N(\varepsilon)}{r_\varepsilon}$ for all $\varepsilon > 0$, where $N(\varepsilon)$ is asymptotically sharp and converges to a finite constant as $\varepsilon \to 0$.
• A sharp sub-Riemannian Bonnet-Myers theorem is proven: if the horizontal Ricci curvature is bounded below by $(2d+2)\kappa$, then the sub-Riemannian diameter is at most $\frac{\pi}{\sqrt{\kappa}}$ for $\kappa > 0$, extending results on contact and 3-Sasakian manifolds.
• The Hessian comparison theorem holds uniformly: $\mathrm{Hess}^D(r)(X,X) \leq \int_0^r \left( \langle D_{\dot\gamma}V, \hat{D}_{\dot\gamma}V \rangle - \langle R(V,\dot\gamma)\dot\gamma, V \rangle \right) dt$, with equality if $V$ is a Jacobi field.
• The sub-Riemannian limit of the Riemannian comparison estimates yields a finite and sharp bound, resolving the issue of divergence in classical estimates as $\varepsilon \to 0$.
• The $J^2$ condition can be removed in the comparison theorems by using the Hladky connection and the index lemma with a metric-compatible connection having skew-symmetric torsion.
• The canonical variation of the metric induces a family of Riemannian structures whose curvature blows up as $\varepsilon \to 0$, yet the comparison estimates remain uniformly bounded and converge to a meaningful sub-Riemannian limit.