Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

TL;DR

This paper classifies 4-dimensional complete gradient shrinking Ricci solitons with half positive or nonnegative isotropic curvature, providing new curvature estimates and a simplified proof of known classifications, with some new results under additional Ricci tensor assumptions.

Contribution

It introduces new curvature estimates and offers a more direct classification proof for certain Ricci solitons with half isotropic curvature, extending previous results.

Findings

Quadratic curvature lower bounds for noncompact Ricci shrinkers with half PIC.

A new, more direct proof of classification for gradient shrinking Kähler-Ricci solitons with nonnegative isotropic curvature.

Classification of 4D gradient shrinking Ricci solitons with half nonnegative isotropic curvature under maximum principle arguments.

Abstract

In this paper, we investigate the geometry of 4-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (half PIC) or half nonnegative isotropic curvature. Our first main result is a certain form of curvature estimates for such Ricci shrinkers, including a quadratic curvature lower bound estimate for noncompact ones with half PIC. As a consequence, we obtain a new and more direct proof of the classification result, first observed by Li-Ni-Wang [35], for gradient shrinking K\"ahler-Ricci solitons of complex dimension two with nonnegative isotropic curvature. Moreover, based on a strong maximum principle argument, we classify 4-dimensional complete gradient shrinking Ricci solitons with half nonnegative isotropic curvature (except the half PIC case). Finally, we treat the half PIC case under an additional assumption on the Ricci tensor.