[Paper Review] Differential Harnack inequalities for a family of sub-elliptic diffusion equations on Sasakian manifolds

Paul W. Y. Lee|arXiv (Cornell University)|Feb 14, 2013

Geometric Analysis and Curvature Flows7 references3 citations

TL;DR

This paper establishes a novel differential Harnack inequality for sub-elliptic diffusion equations on compact Sasakian manifolds under non-negative horizontal Ricci curvature and curvature-dimension-type conditions. By combining techniques from Li-Yau estimates and moving frame methods, it derives a sharp gradient estimate involving horizontal and vertical gradients, leading to a Harnack inequality with explicit time-dependent bounds.

ABSTRACT

We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.

Motivation & Objective

• To extend Li-Yau type differential Harnack estimates to sub-elliptic diffusion equations on Sasakian manifolds.
• To incorporate drift terms via smooth functions $ U_1 $ and $ U_2 $, generalizing the heat equation case.
• To establish a Harnack inequality for positive solutions of the sub-elliptic diffusion equation with drift under curvature constraints.
• To unify and generalize prior results from [3] and [12] by incorporating both sub-elliptic structure and potential drifts.

Proposed method

• Derives a differential Harnack inequality using a moving frame approach instead of the Bochner formula, enabling control over horizontal and vertical gradient components.
• Introduces a modified function $ f_t = -2\log\rho_t - U_1 $ to absorb drift terms and simplify the evolution equation.
• Imposes curvature-dimension-type conditions on $ V = \Delta_{\text{hor}}U_1 + \frac{1}{2}|\nabla_{\text{hor}}U_1|^2 - 2U_2 $, with $ V \leq \kappa_1 $ and $ \Delta_{\text{hor}}V + \frac{n^2}{3\kappa_2}\left(1+\frac{3}{n}\right)^2|\nabla_{\text{ver}}V|^2 \leq \kappa_2 $.
• Uses the horizontal Laplacian $ \Delta_{\text{hor}} $ and vertical gradient $ \nabla_{\text{ver}} $ to decompose the Hessian and curvature terms in the estimate.
• Applies the cost function $ c_{s_0,s_1}(x_0,x_1) = \inf \int_{s_0}^{s_1} \left( \frac{1}{2}|\dot{\gamma}|^2 + W(\gamma(s)) \right) ds $ to derive the final Harnack estimate.

Experimental results

Research questions

• RQ1 Can a differential Harnack inequality be established for sub-elliptic diffusion equations with drift on Sasakian manifolds?
• RQ2 How do curvature conditions on the horizontal Ricci curvature and the function $ V $ influence the Harnack estimate?
• RQ3 Can the moving frame method be adapted to sub-elliptic settings to derive gradient estimates without relying on the Bochner formula?
• RQ4 What is the explicit time-dependent bound for the Harnack inequality in the presence of drift and sub-elliptic structure?

Key findings

• The paper proves a differential Harnack inequality of the form: $ \left(1+\frac{3}{n}\right)\dot{f}_t + \frac{1}{2}|\nabla_{\text{hor}}f_t|^2 - V + \left(1+\frac{n}{3}\right)\sqrt{\frac{n}{2\kappa_2}}\tanh(c_2 t)|\nabla_{\text{ver}}f_t|^2 \leq \frac{\kappa_2}{c_2}\coth(c_2 t) + \frac{3\kappa_1}{n} $, where $ c_2 = \frac{1}{n+3}\sqrt{\frac{n\kappa_2}{2}} $.
• The Harnack inequality is derived by integrating the differential estimate along horizontal curves, yielding $ \frac{\rho_{s_1}(x_1)}{\rho_{s_0}(x_0)} \geq \left(\frac{\sinh(c_2 s_1)}{\sinh(c_2 s_0)}\right)^{-(n+3)} \exp\left(-\frac{1}{2}\left(U(x_1)-U(x_0) + \left(1+\frac{3}{n}\right)c_{s_0,s_1}(x_0,x_1)\right)\right) $.
• The result generalizes the classical Li-Yau estimate to sub-elliptic settings with drift, under the condition $ \overline{\textbf{Rc}} \geq 0 $ on a compact Sasakian manifold.
• The curvature-dimension-type conditions on $ V $ and $ \Delta_{\text{hor}}V $ ensure the time-dependent decay and growth rates in the Harnack bound.
• The method avoids the Bochner formula and instead uses a moving frame technique, enabling a more flexible derivation in sub-Riemannian geometry.
• The estimate captures the interplay between horizontal and vertical geometry, with explicit dependence on the vertical gradient through the $ \tanh $ and $ \coth $ terms.