Let M be an n-dimensional complete non-compact Riemannian manifold, dµ = e h (x)dV (x) be the weighted measure and µ be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakry-Émery curvature is bounded from below by Ric m ≥ −(m − 1)K , K ≥ 0, then the bottom of the L 2 µ spectrum λ 1 (M) is bounded bywhich generalizes the theorem of Cheng (Math. Z. 143:289-297, 1975). As a byproduct, we also get a sharp gradient estimate and a Harnack inequality of the solution u to the equation µ u = −λu.
We establish space-only gradient estimates for positive continuous weak solutions to the p-Laplace heat equation on some complete manifolds evolving under the Ricci flow. As applications, we get Harnack inequalities to compare solutions at different points.
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