On the Scalar Curvature of 4-Manifolds

TL;DR

This paper explores the unique relationship between scalar curvature and topology in four-dimensional manifolds, discussing the Yamabe invariant, presenting new results, and highlighting open problems in the field.

Contribution

It provides new insights and results on the scalar curvature of 4-manifolds and discusses the peculiarities of the Yamabe invariant in this dimension.

Findings

Analysis of the Yamabe invariant in 4-manifolds

New results on scalar curvature properties

Identification of open problems in 4-dimensional geometry

Abstract

Dimension four provides a peculiarly idiosyncratic setting for the interplay between scalar curvature and differential topology. Here we will explain some of the peculiarities of the four-dimensional realm via a careful discussion of the Yamabe invariant (or sigma constant). In the process, we will also prove some new results, and point out open problems that continue to represent key challenges in the subject.