In this paper, the author discusses an elliptic‐type gradient estimate for the solution of the time‐dependent Schrödinger equations on noncompact manifolds. As an application, the dimension‐free Harnack inequality and a Liouville‐type theorem for the Schrödinger equation are proved.

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Let (M n+1 , g) be a compact Riemannian manifold with smooth boundary B and nonnegative Bakry-Emery Ricci curvature. In this paper, we use the solvability of some elliptic equations to prove some estimates of the weighted mean curvature and some related rigidity theorems. As their applications, we obtain some lower bound estimate of the first nonzero eigenvalue of the drifting Laplacian acting on functions on B and some corresponding rigidity theorems.

For any complete manifold with nonnegative Bakry-Emery's Ricci curvature, we prove the gradient estimate of L-harmonic function. As application, we use this gradient estimate to deduce the localized version of the Harnack inequality for L-harmonic operator and some Liouville properties of positive or bounded L-harmonic function.