  We are concerned about the coarse and precise aspects of a priori estimates
for Green's function of a regular domain for the Laplacian-Betrami operator on
any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold
through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough
and sharp Euclidean isoperimetric inequalities. Consequently, we are led to
evaluate the critical limit of an induced monotone Green's functional using the
asymptotic behavior of the Lorentz norm deficit of Green's function at the
infinity, as well as the harmonic radius of a regular domain in the Riemannian
manifold with nonnegative Ricci curvature.
