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Two sharp comparison results are derived for three-dimensional completenoncompact manifolds with scalar curvature bounded from below. The first oneconcerns the Green's function. When the scalar curvature is nonnegative, itstates that the rate of decay of an energy quantity over the level set isstrictly less than that of the Euclidean space, unless the manifold itself isisometric to the Euclidean space. The result is in turn converted into a sharparea comparison for the level set of the Green's function when in addition theRicci curvature of the manifold is assumed to be asymptotically nonnegative atinfinity. The second result provides a sharp upper bound of the bottom spectrumin terms of the scalar curvature lower bound, in contrast to the classicalresult of Cheng which involves a Ricci curvature lower bound.

Comparison theorems for three-dimensional manifolds with scalar curvature bound