  Two sharp comparison results are derived for three-dimensional complete
noncompact manifolds with scalar curvature bounded from below. The first 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 second 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.
