Gradient estimates for positive solutions of the Laplacian with drift

Abstract:

Let $M$ be a complete Riemannian manifold of dimension $n$ without boundary and with Ricci curvature bounded below by $-K,$ where $K\geq 0.$ If $b$ is a vector field such that $\Vert b\Vert \leq \gamma$ and $\nabla b\leq K_{*}$ on $M,$ for some nonnegative constants $\gamma$ and $K_{*},$ then we show that any positive $\mathcal {C}^{\infty }(M)$ solution of the equation $\Delta u(x)+(b(x)|\nabla u(x))=0$ satisfies the estimate \[ {\frac {{\Vert \nabla u\Vert }^2}{u^2}}\leq \frac {n(K+K_{*})}w+\frac {{\gamma }^2}{w(1-w)}\;, \] on $M$, for all $w \in (0,1).$ In particular, for the case when $K=K_{*}=0,$ this estimate is advantageous for small values of $\Vert b\Vert$ and when $b\equiv 0$ it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201–228).