Cheng–Yau gradient estimate

Remarks

The proof applies the Bochner formula to $w = \abs{\nabla \log u}^2$. With the lower bound on Ricci curvature, it gives a lower bound for $\Delta w$ in which $w^2$ appears with a positive coefficient, up to terms in $\nabla w$ and $K w$. One multiplies $w$ by a cutoff function of the distance to $x_0$ and applies the maximum principle at an interior maximum. The Laplacian comparison theorem controls the Laplacian of the distance function in this step. Example: if $\operatorname{Ric} \ge 0$, take $K = 0$ and let $R \to \infty$. Then $\abs{\nabla u} = 0$ for every positive harmonic function $u$ on $M$, which is Yau's Liouville theorem: positive harmonic functions on a complete manifold with nonnegative Ricci curvature are constant. Integrating the estimate along geodesics gives a Harnack inequality: $\sup_{B(x_0, R)} u \le e^{C_n (2 + 2\sqrt{K} R)} \inf_{B(x_0, R)} u$. The term $\sqrt{K}$ cannot be dropped: on the hyperbolic upper half-plane, where $\operatorname{Ric} = -g$, the positive harmonic function $u = y$ has $\abs{\nabla u}/u = 1$ everywhere.

Going deeper

For a student

The estimate is a geometric form of the gradient estimates for harmonic functions in $\R^n$ and of Harnack's inequality. Its proof, through the Bochner formula and a maximum principle, became a model for many later estimates, such as the Li–Yau estimate for positive solutions of the heat equation. It gives Liouville theorems, Harnack inequalities and control of the growth of harmonic functions in terms of curvature.