Ricci flow and PIC1

TL;DR

This paper surveys problems related to Riemannian manifolds with various positive curvature conditions, emphasizing PIC1 as a sharp notion, and discusses how Ricci flow theory effectively addresses these issues.

Contribution

It introduces PIC1 as a precise positive curvature condition and demonstrates the applicability of Ricci flow theory to solve related geometric problems.

Findings

PIC1 is identified as a sharp positive curvature condition.

Ricci flow theory is well-suited to address problems in positive curvature geometry.

The survey highlights recent advances connecting Ricci flow with curvature conditions.

Abstract

We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider. Finally we explain how recent Ricci flow theory is particularly well adapted to solve these problems.