Constant scalar curvature metrics with isolated singularities

TL;DR

This paper proves the existence of complete constant positive scalar curvature metrics with isolated singularities on spheres minus submanifolds, using a more direct method that offers deeper geometric insights.

Contribution

It extends previous results by providing a more direct proof of solutions with isolated singularities and analyzes their geometric properties within moduli spaces.

Findings

Existence of solutions with isolated singularities on spheres minus submanifolds.

Solutions are smooth points in the moduli space when singularities are discrete.

The method offers more geometric information compared to previous proofs.

Abstract

We extend the results and methods of \cite{MP} to prove the existence of constant positive scalar curvature metrics $g$ which are complete and conformal to the standard metric on $S^N \setminus \Lambda$, where $\Lambda$ is a disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen \cite{S}, but the proof we give here, based on the techniques of \cite{MP}, is more direct, and provides more information about their geometry. When $\Lambda$ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2}