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In this paper, inspired by [4, 9], we prove a Sobolev inequality on manifolds with density and asymptotically nonnegative Bakry-\'Emery Ricci curvature., Comment: All comments welcome!

We derive some elliptic differential inequalities from the Weitzenb\"ock formulas for the traceless Ricci tensor of a K\"ahler manifold with constant scalar curvature and the Bochner tensor of a K\"ahler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize K\"ahler-Einstein manifolds among K\"ahler manifolds with constant scalar curvature, and others are given to characterize complex space forms among K\"ahler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among K\"ahler manifolds with constant scalar curvature., Comment: 23 pages

We derive some elliptic differential inequalities from the Weitzenb��ck formulas for the traceless Ricci tensor of a K��hler manifold with constant scalar curvature and the Bochner tensor of a K��hler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize K��hler-Einstein manifolds among K��hler manifolds with constant scalar curvature, and others are given to characterize complex space forms among K��hler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among K��hler manifolds with constant scalar curvature., 23 pages