Gradient estimates for positive solutions of the Laplacian with drift

Let M be a complete Riemannian manifold of dimension n without boundary and with Ricci curvature bounded below by -K, where K greater than or equal to 0. If b is a vector field such that parallel to b parallel to less than or equal to gamma and del b less than or equal to K-* on M, for some nonnegat ive constants gamma and K-*, then we show that any positive C-infinity(M) s olution of the equation Delta u(x) + (b(x)\del u(x)) = 0 satisfies the esti mate parallel to del u parallel to(2)/u(2) less than or equal to n(K + K-*)/w gamma(2)/w(1 - w), on M, for all w is an element of (0, 1). In particular, for the case when K = K-* = 0, this estimate is advantageous for small values of parallel to b parallel to and when b = 0 it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).