Horizontal Laplacian Comparison Theorems

• Horizontal Laplacian comparison theorems are results establishing sharp upper bounds for the sub-Riemannian Laplacian of the distance function under specific geometric constraints.
• They leverage curvature-dimension inequalities, index-form techniques, and Riccati ODE comparisons to convert curvature and torsion bounds into explicit Laplacian estimates.
• These bounds have practical implications for compactness, stochastic completeness, heat kernel estimates, and regularization in Sasakian, contact, and Carnot-type structures.

Horizontal Laplacian comparison theorems establish sharp upper bounds for the sub-Riemannian Laplacian of the distance function in the context of manifolds equipped with a distinguished horizontal distribution and compatible geometric structures. These results generalize classical Riemannian Laplacian comparison theorems, providing crucial analytic and geometric control for sub-Riemannian and foliated settings, including Sasakian, contact, and Carnot-type structures. The core methodology leverages curvature-dimension inequalities, refined index-form techniques, and matrix ODE Riccati comparisons, which translate curvature and torsion constraints into explicit Laplacian bounds. These theorems have profound implications for compactness, stochastic completeness, heat kernel estimates, and regularization properties.

1. Formal Setup and Definitions

Horizontal Laplacian comparison theorems are framed for smooth manifolds $M$ with an orthogonal splitting $TM = \mathcal{H} \oplus \mathcal{V}$, where $\mathcal{H}$ is the bracket-generating horizontal bundle (contact, Sasakian, Carnot, or general Riemannian foliation) (Baudoin et al., 5 Jan 2026, Baudoin et al., 2017, Baudoin, 16 Sep 2025, Lee et al., 2013, Agrachev et al., 2011). The horizontal Laplacian $\Delta_\mathcal{H}$ is the divergence taken with respect to the canonical sub-Riemannian (Popp) measure of the horizontal gradient $\nabla_\mathcal{H} f$. The horizontal distance function $r(x)$ from a base point is defined as

$r(x) = d_\mathcal{H}(o, x),$

measured as the infimum of lengths of horizontal curves joining $o$ to $x$.

Curvature enters through Tanaka–Webster or Bott connections, generating:

• Horizontal Ricci and CR-sectional curvatures (Sasakian/contact)
• Curvature–torsion tensors (Riemannian foliations)
• Torsion and mean curvature invariants (foliation theory)

Under Sasakian structure, $H = \ker \eta$ has rank $2n$, and the Tanaka–Webster connection $\nabla^*$ with curvature tensor $\mathrm{Rm}^*$ is used to bound sectional curvatures and Ricci directions (Lee et al., 2013). Similar adapted connections (Hladky, Bott) are used for general foliations (Baudoin, 16 Sep 2025, Baudoin et al., 2017).

2. The Horizontal Laplacian Comparison Theorems

The essence of these theorems is that, under uniform lower bounds on curvature and controlled torsion/mean curvature, one has

$$\Delta_\mathcal{H}\, r(x) \leq \text{Model}(r(x),\, \kappa,\, N)$$

for all $x$ away from the cut locus, where the model function depends on geometry and effective curvature/torsion bounds.

Sasakian/Contact Manifolds

Suppose $(M, g, \eta, \xi, \Phi)$ is a complete Sasakian manifold of dimension $2n+1$ with Tanaka–Webster curvature bounds:

• $\forall$ horizontal $v$: $$(\mathrm{Rm}^*(\Phi v, v)v, \Phi v) \geq \kappa |v|^4$$
• For the orthogonal complement: $$\sum (\mathrm{Rm}^*(w_i, v)v, w_i) \geq (2n-2)\kappa |v|^2$$

Then for $x \notin \{o\} \cup \mathrm{Cut}_\mathcal{H}(o)$,

$$\Delta_\mathcal{H} r(x) \leq (2n-1) F_\kappa(r(x)),$$

where

$$F_\kappa(t) := \begin{cases} \sqrt{\kappa}\,\cot(\sqrt{\kappa}\, t) & \kappa>0 \ 1/t & \kappa=0 \ \sqrt{|\kappa|}\,\coth(\sqrt{|\kappa|} t) & \kappa<0 \end{cases}$$

Equality is attained in the Heisenberg ($\kappa=0$) and Sasakian sphere ($\kappa=1$) models (Lee et al., 2013, Agrachev et al., 2011).

General Riemannian Foliations

Let $(M^{n+m},g)$ have a splitting $TM = \mathcal{H} \oplus \mathcal{V}$, possibly with non-minimal leaves and non-bundle-like metric. The main tensor is

$$\mathfrak{R}(X,X) - \lambda\,\iota(X)^2 \geq K\,|X|^2$$

for constants $\lambda > 0$, $K \in \mathbb{R}$, where $\mathfrak{R}$ collects Ricci, torsion, and mean curvature, and $\iota$ encodes torsion traces.

The Laplacian comparison reads

$$\Delta_\mathcal{H} r_p(x) \leq \begin{cases} \sqrt{N K} \, \cot(\sqrt{K/N}\, r_p(x)), & K>0 \ N / r_p(x), & K=0 \ \sqrt{N |K|}\, \coth(\sqrt{|K|/N}\, r_p(x)), & K<0 \end{cases}$$

where $N = \frac{n(1+\lambda)}{\lambda}$ (Baudoin et al., 5 Jan 2026, Baudoin, 16 Sep 2025).

Riemannian Approximation and Sub-Riemannian Limits

On totally geodesic foliations with bundle-like metrics, the Laplacian comparison is initially established for the Riemannian Laplacian $\Delta_{H,\varepsilon}$ under canonical variations $$g_\varepsilon = g_\mathcal{H} \oplus \frac{1}{\varepsilon} g_\mathcal{V}$$, then shown to pass to the sub-Riemannian limit $\varepsilon \to 0$ (Baudoin et al., 2017). Model functions are given by the curvature bounds on $K_{\mathcal{H}}(X \wedge J X)$ and $\mathrm{Ric}_{\mathcal{H}}$.

3. Methodologies and Technical Proof Structure

Index-form and Second Variation

Critically, the sharpness of Laplacian comparison stems from optimized second variation (index form) arguments along horizontal geodesics, leading to ODE/PDE comparison results. The horizontal Hessian trace is controlled by optimizing variational fields against curvature ODEs: $u''(t) + K(t) u(t) \geq 0$ subject to boundary conditions, with the optimal Laplacian bound expressed via model solutions $F_\kappa$ or variants thereof. In Sasakian/contact cases, the Riccati ODE for the Hessian block reduces via symmetry (Royden-type comparison) (Lee et al., 2013). In foliated contexts, adapted connections and associated Bochner formulas (including torsion and mean curvature) generate generalized curvature dimension inequalities (CD) (Baudoin et al., 5 Jan 2026, Baudoin, 16 Sep 2025).

Riccati Equation and Matrix ODE

For $r(x)=d_{\mathcal{H}}(o,x)$, the horizontal Hessian $S(t)$ along a geodesic satisfies

$S' + S C_2 S + C_1^T S + S C_1 - R(t) = 0$

where $R(t)$ derives from Tanaka–Webster or Bott curvatures. The Laplacian is obtained by tracing $C_2 S(r)$. Comparison with constant curvature/constant model spaces uses Riccati comparison lemmas to deduce sharp bounds (Lee et al., 2013, Agrachev et al., 2011).

Curvature-Dimension Inequalities

Bochner identities for horizontal Laplacians lead to generalized curvature-dimension bounds: $$\Gamma_2^{\mathcal{H}}(f) + \nu\, \Gamma_2^{\mathcal{V}}(f) \geq \frac{1}{N} (\Delta_\mathcal{H} f)^2 + K\, |\nabla f|^2 + \text{torsion terms}$$ where optimal choice of parameters allows reduction to the Laplacian comparison ODE (Baudoin et al., 5 Jan 2026).

4. Model Spaces and Cases of Equality

Canonical model spaces in the horizontal Laplacian comparison include:

• Heisenberg group: $K=0$, sharp bound $\Delta_H r = C/r$ matches exactly (Lee et al., 2013, Agrachev et al., 2011, Baudoin, 16 Sep 2025, Baudoin et al., 2017)
• Sasakian spheres: constant holomorphic sectional curvature, bound $\Delta_H r = C \cot r$ realized (Lee et al., 2013)
• Carnot groups of arbitrary step: $\mathfrak{R} \equiv 0$, torsion present, but Laplacian bound is linear in $r^{-1}$ (Baudoin et al., 5 Jan 2026, Baudoin, 16 Sep 2025)

A tabular summary:

Model Space	Effective Curvature $K$	Horizontal Laplacian Bound
Heisenberg group	$0$	$\Delta_H r = \frac{C}{r}$
Sasakian sphere	$K>0$	$\Delta_H r = C \cot(\sqrt{K/n} r)$
Carnot group	$0$	$\Delta_H r = N/r$

5. Geometric and Analytic Consequences

Bonnet–Myers Type Compactness

Under positive curvature bounds, Laplacian comparison implies diameter bounds: $\mathrm{diam}(M) \leq \pi\, \sqrt{\frac{N}{K}}$ ensuring compactness and finite volume under canonical measure (Baudoin et al., 5 Jan 2026, Baudoin, 16 Sep 2025, Lee et al., 2013).

Stochastic Completeness and Heat Kernel Bounds

The Laplacian control yields non-explosion for horizontal Brownian motion, via comparison SDE drift terms derived from the comparison functions. On-diagonal heat kernel lower bounds are deduced, e.g.,

$$p_t(x,x) \gtrsim \frac{1}{\mu(B(x, c\sqrt{t}))}, \quad (K=0)$$

(Baudoin, 16 Sep 2025).

Regularization Properties

Gradient and Lipschitz regularization are guaranteed under Laplacian comparison, with explicit exponential decay for the semigroup under positive curvature: $\mathrm{Lip}(P_t f) \leq e^{-K t}\,\mathrm{Lip}(f)$ with sharper gradient-flow inequalities in totally geodesic leaf cases (Baudoin, 16 Sep 2025).

Vertical Laplacian Comparisons

Analogous comparison theorems are available for the vertical Laplacian $\Delta_V$, using variation fields tangent to leaves (Baudoin et al., 2017). These control fine properties for the entire bundle structure.

6. Further Developments and Extensions

Horizontal Laplacian comparison theorems have been generalized to settings with non-minimal leaves, non-bundle-like metrics, and higher-step Carnot groups, utilizing Bochner–Bakry–Émery calculus and curvature dimension frameworks (Baudoin et al., 5 Jan 2026). Quantitative sharpness is achieved in canonical sub-Riemannian models, and the methodology applies to coupled Laplacian operators and two-point distance estimates (Baudoin, 16 Sep 2025). A plausible implication is that further refinement of torsion and mean curvature contributions may yield even sharper analytic inequalities and extend the reach of comparison theory to more general sub-Riemannian and metric measure spaces.

Recent works unify classical Riemannian, sub-Riemannian, and foliation-based Laplacian comparison through effective geometric parameterizations, making these results central for modern geometric analysis, optimal transport, and stochastic processes on singular and non-integrable spaces (Baudoin et al., 5 Jan 2026, Baudoin et al., 2017, Baudoin, 16 Sep 2025, Lee et al., 2013, Agrachev et al., 2011).