Positive isotropic curvature and self-duality in dimension 4

TL;DR

This paper investigates a new curvature positivity condition called half-PIC in 4-dimensional Riemannian manifolds, showing its preservation under Ricci flow and exploring its geometric and topological implications.

Contribution

It introduces the half-PIC condition as a weakening of PIC, proves its invariance under Ricci flow, and analyzes its geometric and topological properties.

Findings

Half-PIC is preserved by Ricci flow.

Half-PIC is maximal among Ricci flow invariant positivity conditions.

The paper explores geometric and topological aspects of half-PIC manifolds.

Abstract

We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: The half-$PIC$ condition. It is a slight weakening of the positive isotropic curvature ($PIC$) condition introduced by M. Micallef and J. Moore. We observe that the half-$PIC$ condition is preserved by the Ricci flow and satisfies a maximality property among all Ricci flow invariant positivity conditions on the curvature of oriented 4-manifolds. We also study some geometric and topological aspects of half-$PIC$ manifolds.