  In this paper we prove that the Sobolev inequality, the logarithmic Sobolev
inequality, the Schr\"odinger heat kernel upper bound, the Faber-Krahn
inequality, the Nash inequality and the Rozenblum-Cwikel-Lieb inequality all
equivalently exist on complete gradient shrinking Ricci solitons. We also
obtain some integral gap theorems for compact shrinking Ricci solitons.
