Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature

TL;DR

The paper investigates 4-dimensional gradient Ricci solitons under nonnegative isotropic curvature (WPIC) and half WPIC, establishing curvature bounds and 2-nonnegativity results for complete ancient solutions, and providing comprehensive classifications for steady and expanding solitons under these curvature conditions. It extends prior shrinker results by showing that weaker eigenvalue conditions ($A_2\ge 0$ or $C_2\ge 0$) suffice to deduce positivity of $A$ or $C$, with detailed implications for 4D Kähler-Ricci shrinkers in complex dimension two. The work unifies geometric decompositions with maximum principle techniques and holonomy arguments to describe all possible 4D steady and expanding models (including products with cigar and Bryant solitons, Calabi-Yau/Kähler solitons, and various quotients), and refines shrinking soliton classifications. It also discusses the stability of 2-nonnegative Ricci curvature under WPIC-preserving Ricci flow and probes the relation between PIC and uniformly PIC, highlighting open problems and pinching criteria relevant to higher-dimensional Ricci flows. Overall, the results advance the understanding of 4D Ricci solitons as singularity models under isotropic-curvature constraints and provide tools for further geometric classification.

Abstract

This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete ancient solutions with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor Rm by |Rm|\leq R. For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient expanding Ricci solitons with WPIC. Finally, motivated by the recent work [59], we improve our earlier results in [24] on 4D gradient shrinking Ricci solitons with half PIC or half WPIC, and also provide a characterization of complete gradient Kaehler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.

Theorems & Definitions (49)

• Theorem 1.1
• Corollary 1.1
• Remark 1.1
• Theorem 1.2
• Corollary 1.2
• Remark 1.2
• Theorem 1.3
• Remark 1.3
• Theorem 1.4
• Theorem 1.5