Geometric inequalities and rigidity of gradient shrinking Ricci solitons

TL;DR

This paper establishes the equivalence of several fundamental inequalities on complete gradient shrinking Ricci solitons and derives integral gap theorems for compact cases, advancing understanding of their geometric and analytic properties.

Contribution

It proves the equivalence of key inequalities on gradient shrinking Ricci solitons and introduces new integral gap theorems for compact solitons.

Findings

Equivalence of Sobolev, logarithmic Sobolev, heat kernel, Faber-Krahn, Nash, and Rozenblum-Cwikel-Lieb inequalities on complete gradient shrinking Ricci solitons.

Existence of integral gap theorems for compact shrinking Ricci solitons.

Enhanced understanding of the geometric and analytic structure of Ricci solitons.

Abstract

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.