Geometric inequalities and rigidity of gradient shrinking Ricci solitons

Abstract

In this paper we prove that the Sobolev inequality, the logarithmic Sobolev inequality, the Schrödinger 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.

Counting ends on shrinkers

“…Recently, Li and Wang [32] obtained a sharp logarithmic Sobolev inequality, the Sobolev inequality, heat kernel estimates, the no-local-collapsing theorem, the pseudo-locality theorem, etc. on complete shrinkers, which can be further extended to the other geometric inequalities, such as Nash inequalities, Faber-Krahn inequalities and Rozenblum-Cwikel-Lieb inequalities in [43]. For more function theory on shrinkers, the interested readers are referred to [18,33,35,36,40,44,45] and references therein.…”

“…The above inequalities are useful for understanding the geometry and topology for shrinkers; see some recent works [32], [33], [42] and [43]. In the following sections, we will apply them to study the volume growth of shrinkers.…”