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  1. ; ; (, The Journal of Geometric Analysis)
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  2. ; (, The Journal of Geometric Analysis)
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  3. ; (, International Mathematics Research Notices)
    Abstract Two sharp comparison results are derived for 3D complete noncompact manifolds with scalar curvature bounded from below. The 1st one concerns the Green’s function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green’s function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The 2nd result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng, which involves a Ricci curvature lower bound. 
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  4. ; ; (, Transactions of the American Mathematical Society)
    We develop Green’s function estimates for manifolds satisfying a weighted Poincaré inequality together with a compatible lower bound on the Ricci curvature. This estimate is then applied to establish existence and sharp estimates of solutions to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity. 
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  5. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract A variant of Li–Tam theory, which associates to each end of a completeRiemannian manifold a positive solution of a given Schrödinger equation onthe manifold, is developed. It is demonstrated that such positive solutionsmust be of polynomial growth of fixed order under a suitable scalinginvariant Sobolev inequality. Consequently, a finiteness result for the number of endsfollows. In the case when the Sobolev inequality is of particular type, the finiteness resultis proven directly. As an application, an estimate on the number of ends for shrinkinggradient Ricci solitons and submanifolds of Euclidean space is obtained. 
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