Connected sum constructions for constant scalar curvature metrics

TL;DR

This paper introduces a general method for constructing constant scalar curvature metrics by gluing manifolds, enabling new examples with prescribed asymptotics, including cylindrical ends, and solutions on infinite point sets.

Contribution

It provides a simple analytic framework for connected sum constructions of constant scalar curvature metrics, extending previous complex constructions to broader contexts.

Findings

Constructed complete metrics of positive scalar curvature on sphere complements.

Produced metrics with cylindrical ends and prescribed asymptotics.

Extended the construction to solutions on infinite point configurations.

Abstract

We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making `analytic' connected sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen's \cite{S1} well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.