Metrics of positive scalar curvature and generalised Morse functions, part II

TL;DR

This paper extends Gromov-Lawson's surgery technique to construct positive scalar curvature metrics parameterized by generalized Morse functions, including birth-death singularities, enhancing the study of metric spaces on manifolds.

Contribution

It introduces an extension of the surgery technique to families of generalized Morse functions, broadening the tools for analyzing positive scalar curvature metrics.

Findings

Extended surgery technique to generalized Morse functions.

Constructed new families of positive scalar curvature metrics.

Enhanced understanding of moduli spaces of metrics.

Abstract

The surgery technique of Gromov and Lawson may be used to construct families of positive scalar curvature metrics which are parameterised by Morse functions. This has played an important role in the study of the space of metrics of positive scalar curvature on a smooth manifold and its corresponding moduli spaces. In this paper, we extend this technique to work for families of generalised Morse functions, i.e. smooth functions with both Morse and birth-death singularities.