Geometric inequalities and rigidity of gradient shrinking Ricci solitons

TL;DR

This work shows that on complete gradient shrinking Ricci solitons, several fundamental geometric-analytic inequalities—Sobolev, logarithmic Sobolev, Schrödinger heat-kernel upper bounds, Faber-Krahn, Nash, and Rozenblum-Cwikel-Lieb—are equivalent up to universal constants, broadening the understanding of soliton geometry. The authors provide explicit Sobolev constants, connect these inequalities through heat-kernel techniques, and derive sharp integral gap results for curvature tensors on compact shrinkers. By leveraging Bochner-Weitzenböck formulas and curvature inequalities, they obtain rigidity statements: under precise pinching, shrinkers must be isometric to finite quotients of standard spaces, such as spheres or complex projective spaces. The results advance the spectral-geometry toolkit for shrinkers and offer concrete rigidity criteria tied to the Sobolev constant and curvature norms, with notable implications for the geometry of shrinkers in low dimensions ($4\le n\le 8$).

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.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Lemma 2.1